Can I use oxygen tablets or liquid oxygen to oxygenate my water for yeast fermentation in mead?

The basics: just mix it up.

I've made a few 5-gallon batches of mead, 30-some by this point. When mixing together my must at the start of a batch, I used to use just a big steel spoon to stir the heck out of the must and make sure I worked in a lot of air and got things nice and frothy. Nowadays I use a regular immersion blender, which has the added benefit of making it easier to ensure I've gotten the thick honey fully mixed into the thinner water. This mixing process has always been enough to oxygenate the must for happy yeasties. Then I add any nutrients, pitch the yeast, get my original gravity readings, and lock down the lid.

The reasons why

Ciaran Haines mentions that oxygen can lead to vinegar. That's partly right, and sustained oxygenation is an integral part of vinegar and kombucha production. More specifically, you want oxygen at the start of your fermentation process, because the yeast use it in the initial growth spurt within the mash or must, as they're getting the ferment going. Past there, if you're shooting for mead, wine, beer, or something similar, you generally want to avoid oxygen, which is why fermentation rigs always have an air seal with some kind of one-way valve to let out the off-gassed CO₂ (the yeast farts). The reason why you don't want oxygen later in the process is that it can cause chemical changes that alter the flavor, and it also provides respiration possibilities for non-yeast microbes, which can then digest the alcohol into acetic acid (vinegar) or just otherwise spoil your batch. Keeping the oxygen out during later fermentation and aging is such a big deal that folks even sell cartridges of argon and other inert gases to flush out any air from opened containers and protect the precious hooch from oxygen. I've never used such things myself, but then wine is much more acidic and oxygen-reactive than mead.

TL;DR:

Don't overthink it.

Don't over-worry it.

Don't throw in weird chemical additives.

Just mix your must really well.

If you want alcohol, you don't need to oxygenate the water - this will give you vinegar instead. Seperate answer, almost everything designed for a fish tank (like bait buddies) will be fit for human consumption because the fish live in the oxygenated water.

I wouldn't think of these things necessarily food safe. I wouldn't be adding something like this to my beer/mead without knowing what other substances are in it. You are overthinking the oxygenation/aeration process.

Aside from using a small O2 tank and stone; people are making excellent mead by simply agitating the must during transfer and/or shaking the must in the fermentor for 5 minutes. There is plenty of O2 in the headspace for good yeast health. It's just a matter of ensuring enough contact with the must. Shaking or agitation temporarily increases the surface area of the must with that available O2.