I know next to nothing about SharePoint, so maybe this isn't something you can/should do, or maybe it's something completely trivial, I don't know, but we have a custom in-house help desk application at work, and I'm wondering if it can be integrated into our help desk SharePoint site somehow?
I really don't know what's possible with SharePoint, so any ideas or thoughts on this matter would be appreciated.
Anything is possible...but check out the MSDN reference for integrating with SharePoint:
Pretty much anything that can go on a webform can go in a webpart - with obvious complications, but yes it would work. Look into webpart development.
The short answer is yes but the amount of time required to make this work will be directly related to your flexibility / needs. Would you be satisfied with default SharePoint lists / forms? Do you need to retrieve and update data hosted in an external source? Do you really need this integrated with SharePoint or simply hosted under the same URL?
I've found that SharePoint can do anything but the time required to make it meet the needs of a demanding/inflexible business user is sometimes significant.
There is also the issue of doing right or simply making it work. Making it work buys you some time initially but you can easily dig yourself a very deep hole that is difficult to escape. My suggestion is to keep the solution as simple and maintainable as possible.
I would try to stick to the features that SharePoint is already offering you. You can achieve a lot by using them, and enriching them with a few simple workflows.
If you want to add some workflow logic to your solution, then try to avoid the designer workflows, since they have some issues when it comes to deployment(in short: you cant). So even if it looks easier to design them in Designer, you will pay a price later when you want to deploy them to production (You have a staging/development enviroment?)
In general I would also agree with mayos answer