I'm not sure if I'm looking at this correctly, but, if I want to encourage support for a bill stuck in committee, is it inappropriate to email the chair of said committee? Otherwise, should I just email my district representative?
The accepted process for a low commitment level bill that you want to chime on about briefly is as follows:
If you are only going to contact one person, contact your own representative and let them know you are a constituent and identify the bill number and a brief (sentences or two) statement about why you do or do not support it. A non-form letter form is stronger. If that person is on the committee, stop there.
If that person is not on the committee, contact the ranking member of the committee who is of your political party as well (the chair of the committee if your political party holds the majority in it).
Of course, if you have a personal connection to someone else in that house of the legislature, don't hesitate to contact that person as well.
There is a form for electronic submission of comments to Texas House committees.
The purpose of this form is to aid Texas residents in electronically submitting public comments to House committees on measures or matters included on public hearing notices. There may be strict deadlines for submitting comments for a particular measure or matter. You should consult the bottom of the public hearing notice for details.
To answer your question, it is not inappropriate to contact any elected official and they or more likely a staffer from their office will give you the time of day to consider what you say (so long as you are polite about it) and consider your argument (If it's a staffer, they will get the message passed on to the elected official. And you'll more likely get staffers... be especially polite to them, as they are just doing their job and most of the time, people don't call their elected officials to tell them to keep up the good work.
That said, it is the delegation that represents your district that you want to call because it's their job to represent the constituency, which means you.
Well... it's their job to get re-elected (dirty secret of all political leaders, they will vote in a way that will get them re-elected) and there is one way to do that... it means listening to the people you represent who will be deciding on your job performance once every so many years. Most legislatures in the U.S. will have a page where you can plug in your address (typically you only need your ZIP Code, but on the side of caution, be ready to have your full address) and will list the delegation that represents you. In Congress (federal) for example, you will have a return that will list two senators and one representative.
As a general rule, the member of the lower house (Usually called the House of Representatives in the U.S. but check your own State to be Sure) will be more available because they have smaller constituencies and more time to meet and discuss (though they are still busy). The Upper House is more difficult to get a personal chat with your delegation members, because they have a larger constituency. That isn't to say you shouldn't try, but rather to be mindful of.
You asked the question in a manner that makes me expect you want a practical answer. You want nuts-and-bolts and step-by-step to have influence over the political process outside the methods prescribed by the election and representative process laid down in the constitution and founding documents of the US and your state.
But, when I express it like that, maybe you see where I'm going to take this instead.
The prescribed process is that you, and your neighbors, elect a representative. (Or some arrangement of reps, depending on the specifics of the case. Maybe house and senate and governor, etc.) Then, when the next election comes around, if they did what you wanted, you vote for them again. (Assuming they still stand for election.)
The prescribed process is NOT that a small group of unusually loud persons get the attention of your rep and convince him that the loud people shouting at him in his office are typical of the electorate.
Sadly, the existing process has huge swathes of just such activity. Of various kinds and levels of acceptability. From newspaper editorials, to petitions, to street protests, to lobbyists with suitcases of unmarked non-sequential bills.
The result is a seriously complicated arrangement of influences determining what a politician will do while in office. And little, if any, actual ethics or integrity involved on the part of the politico. Depending on how resistant the politico is to bribes, intimidation, blackmail, favors, etc., he may or may not do some fraction of what he promised when stumping for election.
By desiring to "put in your oar" in the process, you are in effect going along with it. By far the better thing would be to notice if he kept his promises, and inform your neighbors of that. Then we will, together, encourage politicians to be honest by rewarding honesty, and de-electing those who are corrupt.
But, then, I am entirely used to disappointment. In my life I have encountered exactly one politician who kept his promises. And he got ousted with his party at the next election.
I expect "big tobacco" (or whatever the latest power block with money is) to exert huge influence, often through various forms of bribes starting with stacks of cash. And I expect politicians to be drawn to that influence and those bribes. And I expect political parties to line up behind the leadership with the most money. And I expect the political process to arrange the laws to make such things easier rather than harder.
Consider, for example, the ease with which a member of Congress can buy stock in a company they know will benefit from legislation about to pass. And with absolutely no fear of any tinge of punishment for insider trading.
So, here is what you want to do. You want to send registered letters to every member of the committee. Registered letters require somebody actually accept them. You want as official letterhead as you can manage. If you have a business office, you should put the full contact info of that office. (Don't involve your employer without permission or you could get fired.) If you are in some community group (Shriners or like that) and you can convince your local chapter, you should get them to sign the letters.
Try to avoid form letters. Committees are usually only a few people. You should be able to make customized letters to each person. If they compare these letters and notice that each one is individual, it will impress them.
When you are writing the letters remember the following.
NEVER make ANY kind of threat. Never mention any sort of consequences to anybody. Never say anything that could conceivably be twisted into a threat. Not even such things as harming election chances next time. Any such threat won't get you any progress on your issue, but it might get you the attention of the three-letter-agency types.
Use phrases like "we are aware" and "we hope that" and "we remember your good work on" and "we were enthusiastic over your promises when." Always "we" indicating a group. Always mentioning the good things about the rep. The carrot, the carrot, the carrot. Never the stick.
A politician with many carrots in front of him will imagine all the sticks you ever need.
If you have a group, drop the groups name in the letter a lot. Groups with names like "Citizens for a Better Tomorrow" are great. Keep them vague and suggesting a lot of members who didn't think very critically before joining. Names that you could not possibly disagree with. "You aren't against a better tomorrow are you?"
Emphasize how enthusiastic you are about the bill passing, and how you will remember the great work of the representative when the bill is passed.
Then sleep the sleep of the innocent, recalling that you have joined a revered occupation with a long history. It is at least as old as the Pharaoh and the adviser who stood beside the throne indicating who could speak to the Great Man.