Short answer: use 'Rallies'.
This is a question about spelling, since you already realize that the plural of Rally has an es sound at the end. Either way, Rallys or Rallies, it will be pronounced the same.
There may be formal rules, but there are no official rules. No one officiates the English language, although some people try to, in vain.
So, ultimately, you would need to use your preferred or favorite style guide (such as Chicago Manual of Style (CMOS), APA, New York Times) or just make up your own rule, or go with what looks better to you. This is especially the case with something so esoteric or specialized as card names of a niche card game.
CMOS Online says the following in the appropriate section:
7.8 Plurals of proper nouns
Names of persons and other capitalized nouns normally form the plural by adding s or es. Rare exceptions, including the last example, are generally listed in Webster’s.
The rare exception is the name Romany which CMOS pluralizes as Romanies. So, frankly, that wasn't much help. CMOS, as far as I can tell, does not give a rule for what we are looking for.
Fowler’s Dictionary of Modern English Usage (4 ed.) has only this:
6 Names, etc., ending in -y.
Proper names ending in -y have plurals in -ys: the two Germanys; three Marys in the class; the Kennedys. Note also lay-bys, stand-bys, the ‘if onlys’, treasurys (= treasury bonds).
So, I think there is no standard rule to cover all cases.
There is the professional baseball team of 125+ years, whose name is the Philadelphia Phillies. This seems to be a case of using "common noun pluralization" (philly > phillies) being incorporated into a proper noun, the name of the team. Perhaps this is the rule; it's hard to find a rule for this (and even if we did, we wouldn't have to agree with it).
This case seems similar to the plural of the noun lady-in-waiting, and I think the plural would be ladies-in-waiting. However, I don't know if this word is used as a proper noun. It would be, if it is used as a title: Ladies-in-waiting.
Going on the previous two examples, it seems Rallies can at least not be wrong. And since I'm a baseball fan, I might have to go with Rallies. :)
Sometimes we native speakers do avoid such quandaries by "writing around" (that is, avoiding) the issue with something like Rally Cards.