Of course, sure. Since you didn’t specify whether you meant film/tv or live theatre, I’ll speak for the latter since I work in it. Now, I’m a music director/pianist so I don’t have to do this myself (our work is more word-of-mouth than auditions), but I also play piano for tons of auditions and all my friends are actors, so I know what they go through.
You can join Equity and get appointments for auditions, so you don’t have to sit in a waiting room all day, and if you have an agent they can help you get those appointments, but before you join that union (which you have to earn by… getting a union gig, in a horrible catch-22) you’re “non-Equity”, which means you’re on your own, sucka.
So you read playbill.com or backstage.com and it lists all the auditions for the week, and you go and you wait behind all the Equity people, sometimes showing up at 6am to sign your name for a 2pm audition slot, but you can’t leave in the meantime because you don’t know that they’ll call you exactly at 2. So you get work done on your computer, read, or silently judge the people around you.
It’s a rough life. Most actors who want to do it professionally strive to find an agent and/or become Equity. But just because you get an agent doesn’t mean you can just sit back and let them do the work for you. They’re representing several (sometimes hundreds) of actors, and really they’re there to help you negotiate if you get an offer, not necessarily to send you out on dozens of auditions. But, they do do that occasionally.
But yes, information like that is readily available on the sites I mentioned, as well as actorsaccess.com. I’ve been an extra in a couple of tv shows (Blue Bloods and Flesh and Bone), and while I am in the musicians’ union I’m definitely not Equity. I got those gigs because I’m a member of those websites and I sent them my headshot and resume. It’s a fun thing to do on the side, and I’m not entirely interested in doing it for a career. So yes, you can do it without an agent, but the natural course of a professional career is going to include getting one, and it’s definitely cause for celebration!