I want to do a second fermentation on a batch I did this past week to clean it up more. My question is, I only have one carboy and the bucket I have is missing the grommet for the airlock. Can I siphon into the bucket, clean the carboy and then siphon back into the carboy, or am I asking for trouble doing this?

I’d say that you’re asking for oxidative trouble with the multiple transfers. the yeast that currently in your carboy should be capable of cleaning up any undesirable fermentation flavors.

By “cleaning up” do you want to get rid of undesirable flavors, get better attenuation or clean up excess yeast?

Clean up yeast

That sounds like entirely to much work, and risk in moving a liquid around. And by what you do describe in your post. You don't have the best environment to be doing that kind of stuff anyways. If what you say is that it was brewed less than a week ago. Sit back. Because, the best thing you can do actually is to just let the yeast in there have some more time to finish up. There are a lot of questions I could ask. But, honestly. Extra time in your fermentation schedule will give you the better solution.

The yeast are presently doing just that. Leave them alone. They know what they're doing.

As others have said, let the yeast finish and drop out of suspension. Time will take care of that. I would not recommend cold crashing (refrigerating) unless you have a means to pressurize/close the fermenter to prevent air (oxygen) suck back. The only practical way to do this with a carboy is to use something like a Cold Crash Guardian or a homegrown equivalent made from a mylar balloon or something similar. But that would need to be installed during active fermentation (while CO2 is still being produced).

Good point on the suck back, I mostly fermented in a Corny keg and closed the vent near the end of primary, so suck back wasn’t an issue. Prior to that, I changed the stopper to one with no hole for an airlock before crashing, the bucket lid did get pulled down down from the vacuum. Is there any reason that doesn’t work? It’s been a long time, but I remember the beer being a lot more clear once I started crashing.

That could work in a completely sealed bucket, if it doesn't leak anywhere. I personally wouldn't try it in a carboy (OP's situation) though. There are not really built for interior pressure being lower than exterior. A glass carboy could break. A plastic carboy could collapse to some extent.

I'm replying to stress the point of time, patience, and letting the original yeast do its thing. Also, avoiding cold crash oxygen suck back.

I had the same thought on the glass carboy, the vacuum might cause it to break, plus you really don’t want to move glass around outside in freezing weather.
In case the point hasn’t been established, patience is your friend. When I first started brewing, the conventional wisdom was to transfer to a secondary to get the beer off of yeast ASAP, but now most people want to go from the fermenter right to the final package either into a keg or bottles with priming sugar added individually to each bottle to avoid oxidation from unnecessary transfers.. I saw a video where some guys go far enough to serve right out of the fermenter so there’s no transfers at all.