Can I sign up for dental insurance anytime?

Mostly no, and the reason is where you buy it. On the Marketplace, dental is tied to a health plan. HealthCare.gov is direct about it: you can't buy a Marketplace dental plan unless you're buying a health plan at the same time [1]. So the sign-up window for dental is the health plan's window.

That window has dates. Open Enrollment is the yearly period from November 1 to January 15, when people can enroll in a Marketplace health plan. Outside it, you may still be able to enroll if you have certain life events, like getting married, having a baby, or losing other health coverage, or based on your estimated household income [2]. Job-based plans may have different windows [2].

Buying it is one thing. Using it is another. Dental comes two ways on the Marketplace: a health plan that includes dental, or a separate dental plan with its own premium [1]. And separate dental plans can have waiting periods before they start to cover services for adults. If you enroll in one with a waiting period, it will not cover services until the waiting period ends, and you still pay the monthly premiums during that time [1]. So signing up today does not mean a crown is covered today.

Children are treated differently. Dental coverage is an essential health benefit for children, so if you are getting health coverage for someone 18 or younger, dental coverage must be available for that child, either inside a health plan or as a separate dental plan. You do not have to buy it. For adults, dental is not an essential health benefit, and health plans do not have to offer it [1]. Dropping it is easier than adding it. If you have a separate dental plan, you can cancel it any time. If your dental is part of your health plan, you cannot remove just the dental [1].

Open the HealthCare.gov dental page and read the warning about waiting periods. Then read the Open Enrollment dates in the glossary. If the plan is from work, read the enrollment part of your benefits guide. What a given plan charges or waits: "Don't know yet" here.