As long as you have a suitable supply of yeast nutrients, you can certainly grow non-beer-based yeast cultures for whatever use you need them for. Saccharomyces yeast cultures need the following for successful growth:
- Assimilable nitrogen. Without a steady supply, yeast can't synthesize protein required for reproduction. Various yeast nutrients can supply this nitrogen.
- Carbohydrate, preferably glucose or maltose (for which brewing yeast have a particular affinity). Sucrose (table sugar) is not directly assimilable by yeast and must first be split into its constituents, glucose and fructose, by a yeast enzyme (invertase). Yeast use carbohydrate as the fuel to carry out cell metabolism, either aerobically or anaerobically (depending on the availability of oxygen).
- Small amounts of lipids, or oxygen dissolved in the fermentation medium which will allow the yeast to synthesize its own sterols and fatty acids, key components in reproducing cell walls.
- Trace elements, such as vitamin B, zinc, calcium and magnesium, often critical for the synthesis of yeast enzymes.
Barley wort (which is what yeast transforms into beer) is rich is almost all of these things, and yeast growth during fermentation creates many times more yeast than the brewer needs for fermenting subsequent batches. Hence the natural connection between the industries of beer and yeast-derived products. But since you're concerned about the contents specifically of beer, there's no reason you can't do this without un-involving the offending ingredients.
However, it's worth noting that without some specialized yeast-growing equipment, any relatively simple method you use is probably going to give you a very small amount of yeast to work with, especially relative to the amount of liquid you'd need to ferment. For example, a five-gallon batch of beer (typical for a homebrewer) would likely yield less than a cup's worth of pure, thick yeast slurry when all is said and done. There are certainly ways to increase the yeast density, but only if nutrients are in ample supply and the byproducts of fermentation (chiefly alcohol) don't overwhelm the yeast's capacity to function and reproduce.
So yes, it can be done at home. Whether or not it's worthwhile is an entirely different question.