<p>Our web team has been asked to build some user interfaces in Sharepoint.  The UI's would primarily be forms that would need to write to a  SQL Server database.</p>

<p>Is ASP.NET the best way to do this?  If so what's the best way to integrate the ASP.NET application into Sharepoint?</p>

You have 4 options:
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1. Build web parts 
2. build user controls , and use smart part to display them.
3. use infopath 
4. host the asp.net in an iframe (page viewer web part).
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I would go with building web parts, however user controls and smart part are probably quicker, and quickest still is infopath, but infopath wont use the page layouts or master pages that you specify.
if the web app has already been developed then youre looking at option 4, if its not already developed then I would suggest 1 or 2 which gives you the flexibility of reusing the controls elsewhere in the site as well as utilising the sharepoint metadata.