You do not say whether you have talked to the preschool director and her classroom teacher about your daughter. If you have not -- do so, now! Involve both the teacher (since she will handle your daughter daily) and the director (since she's the one the teacher would go to if there were issues the teacher felt needed further discussion with you). Tell them what you said here. I'm sure they have had kids like your daughter before. Ask them exactly what the "alternatives" to nap time are and how those would work--get specifics, which may do a lot to calm your fears.
Also, you refer to your daughter as hyperactive, but is that just a "she's really busy and talkative" term or is it an actual diagnosis from a doctor? Tossing that term around with a daycare provider if it's not an actual medical diagnosis could set off their alarms and they may ask you if she's seeing a doctor, if she's on medication, etc. So either way, be prepared to tell them about her lively and noisy personality, but if it's not truly "hyperactive" I'd avoid the term. I've noticed parents who use terms like "hyperactive" or "a little ADD" about their kids when their children are not medically diagnosed with such conditions and I think it muddies things for teachers and other professionals who work with the kids. Just a thought.
Ask the teacher and director what you can do at home the rest of the week to improve your daughter's ability to have a "quiet time" each day -- in fact, I'd give her a quiet time every day at the same time they do naps on preschool days. I would tell her she is not expected to nap, but needs to be in her room and quietly occupying herself without other playmates or TV or anything that makes noise. Keep it short. But it establishes a pattern that carries over from school to home and may reinforce the idea that even if you're not napping, you CAN occupy yourself quietly for a short time. It's an essential skill for kids to learn and by 5 they definitely can get the idea.
You are wise to let her do preschool! It will be good for her to be with other adults whose directions she must follow and to whom she must listen. Vital skills for kindergarten preparation!