Answer:
In the U.S, yes they had the right to vote - however, as with Black, Native Americans and Asian citizens, some states had adopted bogus requirements (such as a literacy test) to deny them the vote. This wasn't outlawed until 1965.
An additional barrier for some Mexican-American voters was the official ballot information - it was generally only provided in English. This wasn't rectified in law until 1975. Non-English voting material and assistance must now be provided where needed.