Hello. This is a very big question that might fall a bit outside the scope of these forums. Speaking in broad terms then, yes, it is possible for an Alexa skill to connect to a MySQL database, so long as said database is internet accessible.
Generally speaking, when a user passes an IntentRequest to your skill's back-end where you need to connect to your MySQL database in order to fulfill the request, you'll need to include your own code to make a connection to your MySQL server and then either put or retrieve the information in question. Once that's done you can return a skill response.
Exactly how you do that is up to you and will depend upon which language your skill's back-end is using. Although I haven't done this myself, I found a couple packages for either Python or Node.js which will allow you to programmatically connect to a MySQL database:
I hope this information at least helped point you in the right direction!