I am a computer repair tech by trade and I have clients all over the state. Some of them I access their computers remotely. Is there a way to run my portable apps remotely on their computers or would I have to download the software on their computer and delete it when I'm done?
Unfortunately you can't run anything remotely on your client's computer. You would have to download what you need them delete it after you are done as you suggest.
It may be worth looking into a cloud storage option like Dropbox, box.com or Ubuntu One - then you could have all your PortableApps set up exactly how you want them, zip each up ready to download, and then all you need is a list of links to each app and you would have your full toolkit.
Otherwise if you are not against actually installing something to the client's computer, install the Dropbox client and move the PortableApps you need in/out of a folder you specifically share for this task. Then when you are done just uninstall Dropbox and make sure nothing is left behind.