Translation:My aunt is the president of a large company.
There are planty of roots that end in vowels (heroo, scii, balai, krii, konstrui...). Usually the fact that the accent falls on the next-to-last syllable makes this a lot less confusing than it seems, and a lot of speakers will put a glottal stop between the two vowels.
Sometimes Esperantists disagree on what word to use for a particular concept, and sometimes words change over time; people used to use "firmo" for firm or company, but for whatever reason "firmao" is a lot more commonly used now.
In Esperanto, as long as you follow the rules for word creation, you are free to create. I believe prezidantino is fine, but keep in mind a couple things: 1) the root is prezidi, so this is actually a participle noun for "one who is presiding," not a root for "president." 2) in modern usage constructions like this are presumably not gendered, so indicating a female in this way probably carries connotations depending on your culture and the political atmosphere. If Hillary Clinton gets elected I am not going to call her the "she-president" even though she would be the first. People might infer that I have a problem with it.