No reason to lie

A few weeks ago I saw someone sharing an antivax message on Facebook, quoting someone that claims that vaccines cause infertility, and his rationale for believing that person was that he must be telling the truth because he'd have nothing to gain by lying. What fallacy would that be? Because people lie without anything to gain from doing so all the time!

It is a non sequitur as well as a unsupported claim. As you point out, lying is motivated by more than gain (e.g., the avoidance of embarrassment, the feeling of importance, etc.).

In legal matters, motive is obviously important. If someone has no motive to commit a crime, they are in a much better position than if they did. In other words, motive can be related to truth. The problem is, even in law, "no motive (or reason)" really means "no KNOWN motive." In this way, this can be similar to the argument from ignorance (i.e., they have no reason to lie therefore they are not lying).

"I believe X is true, because the person who told me so would have no reason to lie", or "X is true, because the person who told me so would have no reason to lie."

Besides the point. The person doesn't have to lie in order to mislead you - for instance, they could be genuinely mistaken, and have fallen for internet misinformation claiming the vaccines cause infertility. Whether someone has a 'reason to lie' or not doesn't affect the truth value of the statement.

The second form ("X is true") would be a non sequitur. I also think amazing familiarity is creeping in, because the person says there is "no reason" for the post's author to lie, which sounds an awful lot like mind-reading, as that's not information one can just obtain out of the blue.

Lying isn't the same thing as being wrong.  Just because you aren't lying does not mean you are actually correct.

Another angle on this would be that his response of, "he'd have nothing to gain from lying" is framing it as lying vs not lying which is a kind of false dilemma that only leaves these two options when in reality there are a lot more. For example, he could just be misinformed, incorrect, or deceived in some way.