What are safe alternatives to using electrical tape? I don't have it right now nor am I sure when I'll be able to get it. Could I just use regular scotch tape over one of the shattered festoon bulbs?
As far as safe, duct tape is conductive in some brands especially when wet. Scotch tape is not conductive but it lacks the dielectric properties especially over a shattered globe to be considered safe. To be safe don't use this string of lamps.
Festoon lighting is normally used outdoors or on construction jobs. With a damaged lamp that gets wet a greater shock hazard is there. It would be better to remove the bulb than to have the broken parts extending from the socket.
A trick to removing a broken bulb is to carve the end of a large potato down to a size that will fit into the bulb base, then jam it hard into that base with the broken glass. It will then stick over the broken parts and give you enough leverage to unscrew the bulb base from the socket. DO THIS WITH THE POWER UNPLUGGED THOUGH!! When you are done, wrap the potato and broken bulb very well in newspaper or plastic before disposing of it, otherwise animals (or scavenger people) may try to eat it.