For a web server like this you need a per CPU license, and no client access licenses.
Depending on the spec of the (virtual) machine you are using, and the size the application database is expected to grow to, in the near future you might find SQL Express Edition suitable. This is free to use, even in a public facing commercial setting, but has the following technical limitations:
- it will only use one physical CPU (though I am told it will use multiple cores on one physical CPU), though this is unlikely to affect you in a VM
- the server will not use more than 1GB of memory, which might affect performance if your dataset and VM are large enough for this to be an issue
- it will not allow any one database to have data files larger than 4Gb in total (though "data files + transaction log files" can breach this 4Gb limit)
- extra services such as full-text-indexing and OLAP are not present
- some other advanced features (like indexed views) are not present, though that are not present in standard edition either
With SQL2005 and SQL2008 there is not longer an extra query rate governor, as was found in the SQL2000 equivalent (MSDE). So essentially you are getting the same base engine as found in Standard Edition, just with the extra CPU/memory limits.
So if you project is small enough and doesn't make use of advanced features. If your project is small but expected to grow, you can start with SQL Express and upgrade later - databases can be freely transported between editions.
Edit: some extra detail that I stumbled over this evening: there are some other small memory handling defaults in Express edition that optimise it for desktop use and may affect performance if your DB use pattern is bursty. See here for details.