If the third prong of a plug of a shop vac broke off, can you still use the vac as is, or is there another way to use it without replacing the plug.
You can use it. But it is NOT RECOMMENDED.
The vast majority of the time, the ground plug does "nothing". But under certain fault conditions, the ground plug is vital for safety. There are certain types of tools that are "double insulated" and do not have a ground plug. But if a device has a ground plug then it is not advisable to use it without the plug. This includes both a problem with the plug, as you described, and also includes using a "3-prong to 2-prong adapter".
This is especially the case with a shop vacuum. Shop vacuums often handle water, metal and other conductive materials. If something should go wrong and the ground plug is not intact and functioning properly, that could lead to serious injury.
Replacing the plug is usually not hard to do. Depending on the type of cord, you might be able to replace the entire cord, or you may cut off the end and attach a new 3-prong plug. Look for a replacement that is "UL Listed".