There's nothing necessarily stopping an indictment, but the likelihood that it results in any sort of conviction are small, in my opinion. It's likely that it would get thrown out for one of several reasons.
For reference, the text of the Logan Act:
Any citizen of the United States, wherever he may be, who, without authority of the United States, directly or indirectly commences or carries on any correspondence or intercourse with any foreign government or any officer or agent thereof, with intent to influence the measures or conduct of any foreign government or of any officer or agent thereof, in relation to any disputes or controversies with the United States, or to defeat the measures of the United States, shall be fined under this title or imprisoned not more than three years, or both.
At first glance, it appears that the Senate letter definitely has the intent to "influence the measures or conduct" of a foreign government.
The problem is that it's debatable whether Congress is "unauthorized" in this sense. As representatives of the US government, they may be totally exempt. The language of the law is vague on this point, and courts historically grant leniency in cases where it's not clear.
It's not clear whether the Act is constitutional, since it hasn't been extensively tried in the courts. Especially since First Amendment protections have strengthened in the last century or so, the overly broad language of the act may get thrown out of court if challenged.
Of course, this would probably require an indictment to be tested. Which brings up:
Desuetude basically means that if a law has been unenforced long enough, it might as well not be there. This is one reason you don't see people arrested for violating those "dumb/crazy" laws you see online.
The last indictment from the Logan Act was over 200 years ago, and there's never been a successful prosecution. Whether this would convince the court to throw it out or not, you can guarantee that it would be brought up by the defense, assuming the above barriers were passed.
I'm not a lawyer. Seriously.