Converting 3x7 to a 1x7. Is it possible with only existing parts?

Yes, it's doable. You'll have terrible gear ratio range but you realize this and don't mind.

Either find a narrow-wide chainring that fits the crank, or leave the middle ring in place. You may need shorter chainring bolts as the existing ones have to be long enough to go through the middle and large rings.

You can remove the shifter but leave the front derailleur in place to act as a chain guide, if you can wind out the lower limit enough or otherwise fix it over the middle ring.

You have a zero cost solution, which is to keep the front derailleur but remove the cable (and shifter if you like). Then use the FD limit screws to fix it in the right position to act as a chain guide.

You don't need a 1x chainring, in fact using one with a 7 speed chain is probably less than ideal, so you're actually better off sticking with an existing chainring. You may not be able to get rid of the big chainring, or you may need to get shorter chainring bolts if you do, depending on how the big and middle rings are mounted.

If the chainrings are in good order, I would recommend servicing the front derailleur and continue to run it as 3x. The reason is that front dérailleurs rust quickly but are often chrome plated and it’s actually easy to remove the rust. If the derailleur is temperamental it is typically solved by replacing the shifter inner cable ($1 on eBay) and readjusting the limit screws.

Yes it will take up some of your time, but it would cost close to nothing and you will probably appreciate the wider gearing on some occasions. Also, as noted by others, you’ll still need the front derailleur as a chain guide so you would still see the rusty thing if you would run 1x.

I've done this, converting a bike from 2x11 to 1x11. I removed the big ring, front derailleur, and the shift cable.

I did need parts, but only shorter chainring bolts. If you don't want to do that, you could leave the big ring in place, or rig some kind of shim or spacer.

I did run into one problem, which was that the chain dropped off the chainring several times an hour. There are many gadgets on the market which try to prevent dropped chains. I ended up getting one of them (a chain guide), and I'm quite happy with the bike. But if you want a minimum-cost solution, Chris H's suggestion -- to use the front derailleur as a guide and adjust the limit screws to position it -- sounds good to me.

Not a problem, just remove the front derailleur. If you leave all of the chainrings on the crank, you can change the gear range manually depending on how much gear you want for a particular ride.

I run a small bicycle hobby shop. I have been doing 21 to 7, and 24 to 8 conversions for as long as I can remember.

The conversion can even be done with a riveted crankset that has no bolts in it. I carefully drill out the rivets or spot weld that hold the 3 chainrings together, throw away the big and small ring, and keep the middle ring. Typically in these one piece setups the crank arm is fastened to the middle in 9 out of 10 cases.

Then remove the front derailleur, shifter and cable. I remove between 4 to 8 links from the chain. I have never had any one complain about their chains jumping off the front sprocket. Sometimes I will change the bottom bracket to improve chainline. This conversion has been fool proof for me and my customers.

I have done it too. 3x7 to a 1x7 added a small chain guide just incase. I feel the chain could be tighter but not sure to just buy a new chain(not sure which is best size for the set up) as upgrading a lot anyway or remove a few links but how many??