You've clearly got the general idea, because you instinctively said members get to train their, and not 'member get to train their' or 'members gets to train'. That's the basis of number agreement: the verbs and pronouns and phrases all have to be singular if the noun to which they refer is singular. If the noun to which they refer is plural, then they all have to be plural.
The longer and more coplicated a sentence gets, of course, the harder it is to keep track of which noun these other parts are referring to.
Your example is nice and simple, though - so simple that both of those options could be correct! It depends on how many abilities each member gets to train. If it is only one, then the first option is correct and the second is incorrect. But if they get to train two or more separate abilities, then the first is wrong and the second is correct. Let me finish those two sentences, to show what I mean:
Members get to train their ability to focus.
Members get to train their abilities to plan meetings and to compile documentation.