Know-How Exchange : Other

It sounds like you need two things:

A) A new host. I have been using bluehost.com and am very happy with both the price and their service.

B) More importantly you will need a web developer/programmer. There are many people who doe this. Better yet is to find out if you can maintain the site yourself, but this can be a steep learning curve. You will need to figure out what language this is in and look for a developer skilled in the language. I would work with someone local to where you are. There are plenty of people looking for work these days.

If this company is going out of business I would back up all the files and databases. This may become difficult to do later.

I hope this helps.
Mike

Though you might own the domain name and have a licence to use the software, you might not own the software itself. That will depend on the contract and whether copyright was transferred.

It may be that you do own a licence to the software and can modify the software as you require. You'll need to find out whether it's fully custom built or whether it's based on some underlying platform and, if this is the case then look for who else provides sites using this software and see if they can do the migration/provide the support.

I'd like a better understanding of your expectations...

What type of functions does your website currently serve?