Given that the Prime Minister's activity that has given rise to discontent in her own party, that any EU deal is subject to an imminent meaningful vote, and that no alternative candidate nor markedly better prospects for a deal resolving the NI backstop are obvious (the EU will hardly allow the UK a unilateral right to disengage unless the NI situation is resolved satisfactorily), what are the reasons why an MP would write to the 1922 Committee targeting the PM, rather than simply wait the extra day or two, and vote against her proposed deal?
Brexit is a submarine made out of cheese. Nobody sane at all thinks there is a good Brexit deal to be had. (Every proposed "good Brexit deal" uniformly assumes they can dictate terms to the EU; that has not proven to be the case).
However, May has been forced (in order to become PM) to pretend there is a good cheese-submarine, and has in fact built a cheese-submarine.
This cheese-submarine is bad. Of course it is, it is a submarine made out of cheese. May has to pretend it isn't bad, because it is the result of 2 years of her leadership.
Everyone else is free to say "that cheese submarine is bad". And they are right. May has to say "no, it is a good cheese submarine", because it is the best cheese submarine one can expect (which isn't very good).
They also say "I can make a better cheese submarine". Which is pretty much a lie; nobody actually thinks they can make a better one. But, it is a lie you really cannot prove to be a lie unless the person making the lie becomes prime minister.
So by stating the clearly obvious "that cheese submarine is bad" and the impossible to disprove "I can make a better one", you position yourself (or your faction) to take over the UK government. At that point, well, you sort of won already. You'll probably try to ship a slightly modified cheese submarine, or maybe just let the cheese submarine sink; that is a problem for another day, after your faction is in control.
Defeating this particular cheese submarine (voting against the brexit deal) does not directly get you control of the UK. In fact, once you do that, you might be obligated to propose a better cheese submarine solution, and that isn't a winning move.
Instead, point out the failures of May's cheese submarine (which are large, obvious and undefendable), and use that to unseat May, then get control yourself. Once you have control, you could even just push forward May's cheese submarine as "the best you can do now that May wasted all that time" or whatever. Or force a hard Brexit, and try to get the country to rally behind you. Or negotiate more time from the EU. Or revoke Brexit, then reinvoke Brexit to get yourself 2 more years of power and cheese-submarine making room.
The important part is, nobody actually has a better cheese submarine plan, but many people do want to have power and be in charge.
Theresa May isn't just being targeted over Brexit, though.
- May called snap elections in 2017, in a bid to strengthen her hand. Instead, she lost seats and had to form a minority government propped up by a confidence & supply agreement with the Democratic Unionist Party (where the Tories had outright control before). There's no question she did considerable political damage to herself at this point.
- Brexit was always ill-defined during the Cameron era and May was not the chief proponent. Boris Johnson, the most outspoken proponent, refused to run for PM, likely because he knew the Brexit process would be messy (i.e. how the Ireland/Northern Ireland border will work)
- May's negotiations over Brexit have not exactly been according to plan. The general idea was to get the UK out from under EU rule but the current deal still leaves some EU power in place. As such, many in May's own party would not vote for it. Since her standing as PM is at stake, May postponed the vote to stave off a no-confidence vote but the political blood is in the water now.
The PM has played this disastrously and is now under fire from at least three sources of opposition:
the so-called "ERG", who believe that her deal is bad because it's not Brexity enough. They claim that a better deal with no backstop or a fake backstop is possible, despite all the evidence to the contrary. However, these people probably put in their letters long ago and there's only about 20 of them.
the "moderate" faction, who believe that it might be possible to get a less bad deal by being less Brexity; the "Norway+" option. This also includes actual Remainers, although at this stage there's still no obvious Remain leadership candidate (Soubry? Hardly.)
the "basic competence" discontents: the government was found to be in contempt of Parliament last week, which was an almost unprecedented condemnation. The government has also started trying to delay the vote on the deal. Cabinet ministers openly brief against their own government and vice versa. The basic ability of the government to function is in question and even Tories are going to get fed up of this after a while.
The deal is not necessarily up for immediate voting, I believe that May could delay it indefinitely because she has control of government "programme motions".
No-one in the Commons is unaware that any other PM would face the same Brexit challenges, to much the same outcome; but this is about a more important long-term principle than that.
To add to others' points, the issue isn't necessarily the details of May's Brexit deal so much as how she went about getting it. Legislatures guard against executives that deny their power. Since taking office, she has on multiple occasions been forced in a ruling to give Parliament or others more of a say in the process, rather than acceding to the usual separation of powers. Even the countless resignations and dismissals factor into this, as an unstable Cabinet composition concentrates any power seized from the legislature. Her government was recently found in contempt for hiding legal advice from them, because of what it had to say about the deal.
Indefinitely postponing a vote she was very likely to lose, by all appearances because it could damage or end her leadership, after she'd tried to avoid allowing that vote in the first place, was probably the last straw for many MPs.
May cancelled the vote on her deal and went to see EU leaders with a view to renegotiating it, which was immediately rejected. She can delay the vote until late January, but that leaves little time to do anything else if it is rejected and few people think that there is anything she can do to rescue it.
Reading some of the letters sent to the 1922 Committee, it seems that most of those MPs are unhappy with May's vision of Brexit and her inability to deliver it. They are concerned that the UK will drift into a bad deal or no deal simply by running out of time, and wish to see someone more to their liking take over.
So in answer to the question, by the time they get a vote on the deal it may be too late.