<p>I've written an Access 2003 application to handle internal things at my company over the past couple years and we are talking about upgrading all of our computers to Office 2007 which means Access will be updated.  Is this going to cause a problem for me?</p>

Allen Browne, Microsoft access MVP, has written a comprehensive article on Microsoft Access 2007 and upgrading:
Converting to Access 2007
Here is an article by Microsoft:
Transitioning Your Existing Access Applications to Access 2007
In case you are not aware, it is possible to
try 2007 for free
. I would not recommend trying on a PC that you use regularly for other Access work, it is not impossible, but things get awkward.
If you wish to run your 2003 version on 2007, make sure that you create an mde, otherwise you will run into problems. Access 2007 treats several controls in a different way, for example, textboxes with date fields get calendars attached. You will also need to check your references. I also recommend that you
self-certify
your projects. Other than that, I have seen posts on a number of problems, but these are usually specific to the poster.