A Large, Fresh Spill
Consider a worst-case scenario: You're painting the wall or the ceiling and the latex paint can slips, dumping a large puddle in the middle of your vinyl floor. In this case, you're lucky that the paint is water-soluble. Start the cleanup by covering the spill with cat litter or sawdust to absorb most of the paint. After picking up the paint-saturated material with a paint scraper or drywall blade and dumping it in a garbage bag, you should be able to clean the rest of the stain with soapy water, as long as you act quickly. Rinse with clear water, and you're done.
Scraping Hardened Paint
An even worse-case scenario than a fresh spill is one that has been on the floor long enough to dry and harden. You have to scrape the bulk of this off before a solvent will be of any benefit, and this creates the danger of scratching or cutting the vinyl. Minimize this damage by using a stiff paint scraper and keeping the handle at a low angle with respect to the floor. An even safer method that is worth trying first is to scrape with a plastic paint scraper, using a hammer to tap into the hardened material and chip it away.
About Latex Paint Remover
If you're cleaning a dried spill, or you manage to get a large spill while it's fresh, but don't act quickly enough to get the residue before it dries, your efforts to remove it with soapy water might not be successful. Before you use latex paint remover to wipe up the dried paint, test it on a small, inconspicuous part of the floor. Some products contain solvents that you find in polychloride vinyl pipe cement, including xylene, and vinyl flooring is essentially a form of PVC. You may find that such products dull the floor finish or eat away at it -- avoid using them.
Safe Latex-Removing Solvents
If you find the brand of latex paint remover you have in your utility cupboard unsuitable, you have other options. One is to wipe the dried residue with denatured alcohol -- which you can buy at a paint store -- or isopropyl alcohol -- which you may have in your medicine cabinet. Another option is to use a citrus-based cleaning solvent or an industrial hand-cleaner, which usually contains one or both of these solvents. Moisten a rag with alcohol or citrus cleaner and lay it on the residue or, if you're using a hand cleaner, spread it on the residue. Wait 10 minutes before wiping it up, then clean the area with soapy water.