If you are talking about the vertical style bread toaster, don't cook anything that isn't bread in those. If you are talking about a toaster oven, you can wrap it in foil and bake it.
A solid alternative, and what makes a hotdog's flavor really pop, is cutting it up into about 10 pieces and pan frying it.
Please don't experiment with cooking anything but bread in the toaster. They're extremely dangerous to fuck around with. Please also keep anything conductive out of a toaster when it's plugged in (even when it's not plunged down) if you use some soft metal like tinfoil there's a chance some gets stuck in the toaster and bridges a section of the heating element.
As for using a fork to fish it out that's generally not actually a problem - just unplug it first and make sure you're not scraping the element in any way (I occasionally toast home made irregular bread so I often use a knife, carefully, to extract it).
Toasters, especially old ones, oftentimes don't have an integrated fuse so you can really fuck shit up with one.
You will almost certainly have to stick a fork down into the vertical style bread toaster to retrieve the hotdog, which is hella unsafe.
It's also almost impossible to clean the vertical style ones, which isn't a problem for bread since the extra bits just burn off, but it is a problem for other foods like meats, as bits can easily rot in there.
Meats can also drip grease onto the elements, which will cause a fire.
I'm not sure what cookware you have, but it sounds like you may be pretty limited on choices/space. I would highly recommend swapping the vertical toaster for a toaster oven. As they are basically little ovens that you can cook almost anything in, including toast and hotdogs.
You sound like you maybe have very little culinary experience, and I would be happy to assist, as this is one of my hobbies (and I have worked in kitchens professionally as well). And my community could sure use more content! 😅
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.
Do you have a microwave? I put lightlife's in my microwave for 90 seconds on rotate and they come out okay. Or you can boil and then pan-fry to make the skin crispier, or you could just pan-fry.
If it has fat in it (it probably does) it can drip out and start a fire. I’d microwave it or boil it before putting it in a toaster. A toaster oven with a pan is fine, if that’s what you mean? Some people call them the same thing.
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.
It's usually the influence of the Darkness that prompts people to put hotdogs in a toaster. Don't listen to what it tells you about the bathtub either.