You state your user base is expanding. Does the current UI suck? If so, what reasons are there for the expansion? Is a bad user experience slowing the expansion, or is it something existing or potential users are talking about? If yes, continue.
If your UI/UX sucks (if it's a green screen then that's another story altogether), pick up a copy of Rocket Surgery Made Easy by Steve Krug and setup your own UX testing internally using existing staff. Don't let the developer of a particular thing handle a session for that thing (too much personal bias, I have been that person before)
Pick a specific use cases to test, if users complaining about something in particular that's what you need to focus on.
Do the first round of tests on internal staff, especially on bits the "customer" isn't familiar with. Learn how UX testing works, it isn't that difficult.
Then follow the book for external UX testing. Remote is pretty easy these days, but in person is better (read the book, pay people and provide snacks and such, make them feel appreciated and also worth their while).
With UX testing results in hand, mock up a prototype to show how the changes impact the user experience. UX with the prototype.
Doing this can go quite quickly for a single use case (and then drive further changes as people's eyes open).
Sell it to management, based on concrete potential/existing user comments as well as before/after (prototype) UX feedback.
Rinse and repeat.