I have a school issued MacBook Air with several apps installed. While they were downloaded from the App Store using my personal Apple ID, I need an administrator password, which I don't have, in order to delete them. Is there a way to do this without adding another account?
No, but you can revoke the computer from your AppleID account.
Unfortunately, this is not possible without bypassing the security systems (which i do not suggest doing.) I'm assuming that you really need these apps to be gone (like having disabled xcode command line tools), so I recommend you put together a good argument for your school's tech helpdesk to uninstall them for you, something along the lines of 'I was an idiot when I installed these, please help'.
If for whatever reason you don't want to do that, there are several good ways to soft-brick your computer to the point that it will need a complete reimage. This is very mean to your school's IT people, but if you back up your files and then get them to reinstall OSX on your computer, that will certainly get rid of the apps, hopefully without too much suspicion.
This is more of a psychological solution than a tech-based one, but I hope it helps!
I don't really have a way that you can delete the apps without the admin password but i have two ways you can change the admin password meaning that you can then delete the apps. My personal favourite is recovery mode because it is easier http://www.macworld.co.uk/how-to/mac/what-do-if-youve-forgotten-your-mac-password-3594395/ MAKE SURE FOR RECOVERY MODE YOU SKIP STEP 3 OR YOU WILL WIPE YOUR MAC
you can by putting it in the trash and then emptying your trash!
https://superuser.com? it will show you