No (most companies won't because of insurance and accountability, if they break something is that your fault or theirs?). Custom pc builders are able to make competitively priced pc's because they bulk buy components so they can only make pc's with those parts. You may find a smaller more local store to do it for you but they will probably charge you more for the service. You'll probably be able to get something similar if you look at custom builders and their options. Your best bet to do it on the cheap is to befriend some local pc geeks and find someone willing to help you (im sure someone wouldn't mind wasting a few hours one weekend building a pc for a few dollars).
Nice budget build by the way. See if you can fit in a 120gb ssd or something for your OS and other frequent programs that you would expect to run on startup (steam, internet browser etc) you won't regret it.
The other option is to learn to build it yourself, it's not as hard as it looks. There's a million videos on youtube that take you through a step by step process on how to build them and is how I learned myself. Watch them until you find yourself going "yeah connect that to that, that to that, yeah yeah, this is boring me now because I know what to do"