How did things work out for you? Still with us?

Don’t put the glizzy down the toaster slot, dawg. It deserves better, and so do you.

Like another comment said, if it's a vertical toaster, probably a bad idea. You can simply just fry it on a pan if you have one. Make cuts along the hotdog and fry it in some oil. It will turn out nice and juicy.

You need to clean out the crumbs tray before you do this, and then wash it after. Make sure that the inner grating does not have anything stuck after too.

Toasters are only intended for dry bread. Inside they are usually just a very simple timer chip and a small 2 piece transformer. When the lever is pulled, part of that transformer is connected using a piece of steel. There is a momentary switch that powers the timer circuit which then connects the transformer holding the steel in place due to the magnetism. When the timer stops powering the transformer, it releases the steel when the magnetic field collapses.

When the circuit is connected, the power through this small transformer is traveling through a nickel-chromium ribbon wire on both sides of the slots for bread. This wire is just wound around some mica sheets that can handle high temperatures. As far as the mica and nicrome wire, this construction style is used in most consumer quality devices. You'll find this in hotplates, hair driers, curling irons, and toasters.

Note that, the transformers used in the timing circuit of toasters are not the isolated type. The nichrome wire is connected to the live mains circuit when powered.

I say all of this to make the statement: fundamentally, it is just a heating element. The fact that the element is inside a form factor is rather irrelevant. The relevant part is that the wiener likely contains fat and liquids that will come out with heat. If these were to squirt onto the mica or nichome wire it would be a problem that must be remedied with a tremendous amount of work or it could leave traces that might cause a fire later. If these juices got into the circuitry, it will likely destroy them and ruin the toaster. The placement of this circuitry is likely somewhere where an issue is unlikely, but all I can do is speculate. The engineers that designed the thing likely were not given a if-wiener-insertion constraint, and I don't think that is a UL listed thing.

Ultimately, you have to play the engineer and determine the cost to risk given the known and potential factors; it is just a simple heating element. This is only slightly more complex of a situation as a cave person's if-campfire wiener insertion conundrum. The plastic molding and appliance form factor aesthetics are irrelevant.

You’ll start a grease fire almost certainly, but sure, go for it. It will definitely be cooked.

I tried microwaving a hot dog recently (I had a microwave for 30 years and never thought to try!) and it works surprisingly well. The skin was even browned.

do you not have like an air fryer or a electric grill/panini press you can use?

It would probably work in a pinch, but do not use the darkness settings, keep a very close eye on it.

A toaster oven? Sure. Use the broil setting and a broil pan or a normal pan, turning occasionally. Broil is just an upside down grill.