Yes it is possible for a woman get trichomoniasis on her own.
You can get trichomoniasis on your own by touching your genitals without washing your hands.
The trichomoniasis can be transmitted through touching surfaces such as a public toilet flush handle, bathroom door, faucet etc.
Most commonly the trichomoniasis is transmitted sexually and you may get it and have the trichomoniasis through a previous relationship.
Some woman can have trichomoniasis for months and not even know about it.
So while the trichomoniasis is usually contracted and transmitted sexually it is possible to get the trichomoniasis even without having sex or being sexually active.