Your question as phrased (Is there some way that I can tell SVN to remove (from the repository) any files that are missing in the working copy?) is really a duplicate of the related question svn command to delete all locally missing files. In response to that question, I provided a one-line PowerShell script to let you automatically identify and remove the requisite files from your repository. I reproduce it here for convenience:
svn status | ? { $_ -match '^!\s+(.*)' } | % { svn rm $Matches[1] }
That assumes, of course, that you are on Windows. Here is an analogous solution if you are on Linux/Unix.
However, when you add in part two of your question, which you inadvertantly revealed :-) in a later comment--Is there a way to make this happen automatically upon a commit?--then your question is unique.
The only avenue I could see to answer part two is using a pre-commit hook, i.e. a hook script executed upon initiating a commit but before the commit itself actually occurs. Normally, you would use a pre-commit hook to validate some set of conditions and either allow the commit to proceed or terminate. Alas, I do not think it is allowable, let alone advisable, to modify the set of files being committed during the commit itself. But I have not found this point documented so you may want to dig further on this point.
2012.06.19 Clarification per Don's comment
Don is correct that you need a client-side hook, not a server-side hook because the deleted files to process are on the client. Command-line svn does not have this option but TortoiseSVN does offer client-side hooks. In fact, after I perused the referenced manual page I was reminded that not only does TortoiseSVN offer a pre-commit hook, but it also offers a start-commit hook, which is ideal for this situation. The difference is that the pre-commit hook runs after you open the commit dialog and after you press OK to attempt to execute the commit. The start-commit hook, on the other hand, runs before the commit dialog opens. Thus, with the hook script in place, by the time you see the list of files to commit it will include the files it is going to delete as well!