I host Plexum software on my own server. There is no csv export in the admin side of the shopping cart software.
I'd like to move to Bigcommerce. Is there any way this can be done being that I have server access to the database and files?
Lol, oh being trapped in eCommerce hell, it's a scary place; only a few have made it out uncorrupted.
To answer your question, yes if you have access to the database then you should (assumingly)(is that a word? you get my point though) have access to all of the required data you need to migrate. Since you have access to the database, you can actually export the database as a CSV and work with it from there - there are plenty of resources on how to do this. Or, you can create a program that loads the data from your database and imports it into BigCommerce via its web API. If this sounds foreign to you, it might be best to contact a professional.
If you ask a more specific question, you'll receive a more specific answer. Until then, best of luck on your journey back to the surface.