Yes, you can apply on your own, even if you currently live with other people.
If you are applying to an affordable housing program on your own, you may apply as a single-person household. Sometimes you will see the word "family" be used, but in regards to housing assistance programs, a "family" can consist of one person.
When writing in the members of the household on your application, do not name the persons you are currently living with. This section is to name persons who will be living in the household while receiving assistance.