I have a client who would like an auction site. I have decided to use Wordpress for this and found proper hosting and what not and thought I was set. Then the client asked if I could save time and money on their part by hosting this event site on their main site as well as their domain. IE: www.123.com/auction. The main site was not created by me and is not powered by WordPress. It is also hosted by some sort of "school community" website platform. Is it even possible, If I can access the file manager in the first place, to host a wordpress database and CMS separately on one hosting account for what is pretty much an entire different web page. Is this even practical?
Assuming that the hosting account is capable of meeting the minimum requirements, yes, you can install WordPress in a subdirectory of an existing site.
See more info here: https://codex.wordpress.org/Giving_WordPress_Its_Own_Directory