Is it common for voice actors to have their own equipment?
Yes, unless you want to pay more than you will get for simple jobs. Transportation + Studio recording fee.
Get a USB microphone https://duckduckgo.com/?q=usb+microphone There are some not too expensive.
Before buying one read some reviews and HEAR some reviews, youtube is a good place for it.
You could start with a cheap one, and if the business is for you, get a more pro mic for cleaner and brighter sound.
Are there sites where you can register as a "voice" actor
Yes... if you are one. But you are not yet, you do not even know if you would like to do that. Like all activities, you can become good at something or just not.
So, first of all, try youtube, publish some video narrating a story, try royalty-free classic stories, (you can not simply buy a book and use it, even if the original copyright has already expired because the version could still have it)
Find noob writers and ask them about recording one story and make a channel.
You also can use a website where people pay a small fee, like Fiverr, record some demos, and see if people are willing to pay you for the quality of work you offer.
Compare prices. Do not offer the cheapest price or you will never know if they want you for the price or for the quality.
If it is a feasible road to you, you need to take courses. Again, there are pro-people and people that just want to play doing stuff. There is a lot to learn in the field. As a talent to be nurtured and as a business.
Do NOT do something because your friends tell you. Do something because you want to!
One edition to make the answer on-topic on this video production site:
Offer your voice as a voice-over talent for documentaries with a small budget.