I currently have (2)2008 Microsoft SBS servers that I am being told cannot be updated without buying new hardware. I am wondering if there is a possibility of using these existing servers to install a Linux operating system on.
I currently need to be able to access these servers remotely while traveling in order to share files and operate programs with other users across a network that has roughly 15 users.

The answer is almost certainly YES.

To really know you'd have to give the hardware specifications.

I verified SBS is just the name of the OS and apps you installed and doesn't include the hardware itself.

You'd have to determine if there were any apps that you're currently using on SBS that you couldn't live without. It wasn't clear if you intend to buy new servers for SBS and only intend to use them for fileservers. If so apps shouldn't be an issue. If not you may find open source equivalents of apps. (e.g. OpenOffice or LibreOffice in place of MS Office.)

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What I would like to also explore as an interim solution is to leave all of the programs that I have loaded on the MS Small Business Servers and possibly just use a Linux server to access the internet and use for browsing. Do you think that is possible?

Do those servers need to constantly available for those other users ?.
If so you are looking at running Linux as a virtualised guest. Certainly do-able. Virtualbox should run on SBS as it looks like it is 64-bit Windows Server 2008 based. Simple enough to install VBox if you have admin rights I would think, then just follow the manual.
If you can shut down SBS, Just about any Linux should install if you can free up some disk space - this is called "dual boot". You either boot SBS, or Linu . Not both at once.
To see if everything would work, you can download a Linux image and burn it onto a USB or CD/DVD and run that. It will run without updating your system at all, but you can check network, video card, memory requirements, see if you can get onto the web ...
Known as liveCD, and all Linux distributions allow this.

Windows already has a good way to access remotely. https://technet.microsoft.com/en-us/...(v=ws.10).aspx

You should be able to easily install linux but the complexity of it may be not worth it to new users.

Yes - you could certainly setup a separate system with Linux to access the internet. Note that the browser most Linux distributions use is Firefox by default. You can install Firefox on Windows as well.

There are ways to do Linux on the same system as Windows:
1) Dual boot - In this setup you have a partition used for Windows and setup a separate one for Linux. To access one or the other you reboot into that OS. (Probably NOT what you would want to do on your SBS if you want to keep SBS but is a possibility.

2) Install Cygwin in Windows. This is a Linux distribution that is optimized to run alongside Windows. It has some quirks that a straight Linux installation doesn't but works for many purposes.

3) Install WINE in Linux. WINE is designed to allow you to run Windows apps inside Linux (it requires a Windows license as you're actually installing Windows inside it). Using this you can use things like IE that unique to Windows but you have to research specific apps to see if they run well from WINE.