TL;DR
Yes, in moderation. Vinegar can cut odors and light mineral buildup, but daily use may wear down rubber gaskets over time.
Vinegar shows up constantly in laundry hacks: deodorizing towels, brightening whites, softening fabric. Most of that reputation is earned, but occasionally and every single load are very different habits.
The acidity that makes vinegar useful against odor and mineral deposits is the same property that can gradually affect the rubber seals and hoses inside a washing machine with heavy, repeated use.
What Vinegar Actually Does in a Wash Cycle
Plain white vinegar, about 5% acetic acid, can help neutralize odors trapped in towels or gym clothes and dissolve light mineral deposits from hard water. It's not a replacement for detergent, it doesn't lift oil-based stains or dirt the way surfactants in detergent do.
How to Add It Correctly
About half a cup added to the rinse cycle, or poured into the fabric softener dispenser if your machine has one, is enough. Adding it to the same cycle as detergent isn't harmful, but keeping the two separate lets the vinegar work during rinse without competing with the detergent's cleaning action.
Cleaning the Machine Itself
Running an empty hot-water cycle with 2 cups of white vinegar every month or two can help dissolve detergent buildup and mineral scale inside the drum and hoses, which is a different use than adding it to laundry loads.
The One Combination to Avoid
Never combine vinegar with bleach, in the machine or anywhere else. Vinegar is acidic, and mixing it with chlorine bleach releases chlorine gas, which is dangerous to breathe even in small amounts. If a load needs both odor control and disinfecting, run them as completely separate cycles, never together.
A splash of vinegar now and then is a reasonable, low-cost laundry helper for odor and hard water buildup. Just check your washer's manual before making it a permanent habit, and never let it cross paths with bleach.