I was riding my 90's Trek 200 the other day and the RD snapped in 2 like a twig (because a twig became lodged in it while riding in more mountain bikey, off road conditions than I should have been) . The FD had already broken beyond repair last fall. I was hoping to just take off the RD and shorten the chain so that it will stay on one gear, but when I do that, the chain just skips up to bigger and bigger gears on the rear cassette. Clearly this conversion is a little more difficult.

The bike has vertical rear dropouts, if that's important

I was hoping for this to be a cheap/free fix for riding my bike around campus cheaply. Can I just get a chain whip and remove the rear chainrings and put the one I want back on with a spacer? Does it really have to be complicated?

Depends how good you want the results to be.
But yes, for a cassette freehub, generally just a bunch of spacers with one cog will get your chain line correct and stop the shifting problems.
With vertical dropouts you'll have some chain tensioning issues that will require experimentation with different cogs and maybe a half link.
Check the Sheldon Brown pages for conversions.

Yeah, everything's always more complicated than your first glance.

1. First of all, you're talking about a single speed. A fixed gear means that you can't coast.

2. You have two issues to resolve - chain line and chain slack.

Chain line is easy. The rear cog that you use has to line up with your front chainring. There's two different possibilities. If you really have a cassette, it's possible to remove all of the cogs and then replace all but the one you want to keep with spacers. The trick is how many spacers you use on each side of the cog so that it lines up perfectly with your front chainring. If you have a freewheel (my bet) the cogs are attached to a separate ratchet mechanism. Remove that and replace it with a single speed freewheel. Then you have to figure out how to spacer the front chainring to match the plane of the freewheel.

Chain slack take up. If you don't keep your chain pretty tight it will slap around a lot and occasionally jump off it's sprockets. Since your bike has vertical dropouts you can't simply pull the rear wheel back until the chain slack is tight. A complicating factor is that generally you can only adjust chain length in 2" increments. The easiest way to do that is to use a derailleur to take up the chain slack. There are other gizmos you can buy that do the same thing. A second method is to use a half link. If you look at a bicycle chain you'll see that it has wide links and narrow links. A half link is wide on one end and narrow on the other. It's advantage is that it allows you to adjust your chain length by a single inch. A third method is called a "magic gear". That's a combination of rear cog and front chainwheel that happen to require the exact length of chain that fits between your crankset and vertical dropout.

That's the short version. The devil is in the details and there's lots of details to everything that I've said.

a spring wrap up or a fixed tensioning pulley for chain tension
will let you convert to a singlespeed freewheel bike.

If you spring for a us made eccentric axle hub, then there is a possibility of a fixie or a single speed.
on a vertical dropout frame.

whole section on that stuff.. https://www.bikeforums.net/forumdisplay.php/178-Singlespeed-amp-Fixed-Gear
it's so trendy, it's a crowded marketplace.
you can just get one finished and brand new
for the cost of the conversion parts, or less.

me: I need more than 1 gear ratio..
My Internal gear hub bikes, get more riding than the derailleur bikes..

https://www.nashbar.com/bikes/Product...1_10000_201527

Nashbar sells a conversion kit for $20

Oops, I left that one out. They're big money and although I've never tried one myself, I understand they don't work all that well.

Including a new rear wheel? killer deal !

having a symmetrically spoked balanced tension wheel ,
rather than one so dished to make room for a 9 speed freehub,

is the 1st benefit of having a single speed on the rear wheel..

or an IG hub..

https://www.sheldonbrown.com/free-k7.html

https://sheldonbrown.com/singlespeed.html

These should get you started. Yes, there's more to it than what you've done. If you want to do this as cheaply as possible get an old rear derailer and just dial up the limit screw to whatever gear you want to stay in. It won't look great but it will work.

1"

Unless you use a half-link. Then you can get to 1/2" increments.

All in all it would be cheaper to buy a new low cost derailleur.....

Not saying you should. Just that if you have to buy ANYTHING at all to make this conversion work that likely you may as well just buy a cheap new derailleur.