My mothers family came over in 1961 and I have recently begun a journey to try and locate information. I was hoping to locate something similar to what Ellis Island has but it seems that does not exist. I have requests in with the Miami Tribune and the Miami Dade College but haven't had much luck with them. Any help would be greatly appreciated. Apologies if this is not the right subreddit.. If I should be asking somewhere else please let me know. Thank you!
It sounds like your main interest is in your mother's history, not a database for research purposes.
One important tool would be a FOIA request. Your mother (or you, in case your mother has passed) can request her complete A file (alien file) from the government. In this case, USCIS which is subordinate to DHS. That is free unless they find something, in which case they charge depending on the size of the file (usually under 100 dollars).
You might also want to contact USCIS directly to ask them if such a database exists already.
Finally, as dumb as it sounds, try ancestry.com . So long as you have your mother's full name you can find out if ancestry's agents have found any certificates under her name in the public records of the State of Florida or elsewhere. My grandmother popped up there and it gave me a couple of leads.
Oh, also try to contact specialists in Cuban immigration (there is a professor at DuPaul U in IL, for example. They may offer more suggestions.
Good luck.