There is no level restriction. You can become a vampire right now if you can do one of the following:
Go to a specific spot in Bangkorai, Reapers March, or the Rift and try to get bitten by special bloodfiend enemies.
Find a vampire player who has a bite available, or
Buy it from the Crown Store.
3 is the fastest and easiest, but it costs real money and you do miss the vampire introduction quest. The recommended method is #2. For that, simply ask zone chat if anyone has a vampire bite available (I suggest doing this in one of the three zones mentioned in #1 above). The ability to pass on vampirism is a bit later in the skill line and does have a week-long cooldown, so not every vampire you see can do it, but there are a lot of older vampires around at this point, so chances are you'll find someone if you just ask politely.
Living vicariously through my characters.
My Girls:
"If you were trapped in your house for, say, a year, how would you pass the time?"
Nephikah the Houseless, dunmer assassin: "I suppose I could use the break. I have a lot of business holdings now that need management." Swum-Many-Waters, elderly argonian healer: "I think that I would enjoy writing a memoir." Silh'ki, khajiit warrior-chef: "Would this one be able to go outside, to the nearby river? It's hard to fish without water!" Peregrine Huntress, bosmer hunter: "Who is forcing me to stay inside, and where can I find them?" Lorenyawe, altmer mechanist: "And why would I want to go outside in the first place? Too much to be done in the workshop." Lorelai Magpie, breton master thief: "I'd go nuts. Lucky for me, I have a little experience sneaking out!" Rasheda the Burning Heart, redguard knight: "I would continue my training to keep my skills sharp." Hex-Eye Azabi, khajiit daedric priestess: "I suppose it would be lucky, then, that I built a shrine to Mephala in my backyard." Yngva Stormhammer, nord bandit (reformed...ish): "I hate being inside even when I'm not forced to be. GET. ME. OUT." Madam Argentia, vampire dunmer aristocrat: "I suppose it would be more of the same. I have a rather... contentious relationship with the sun." Mazie gra-Bolga, orc scout: "Uh... I'd have to house train my bear..." Felicia the Wanderer, imperial witch-for-hire: "What Lorelai said." Calico Jaka-dra, retired khajiit pirate: "This one would like a rest from her grand adventures. Her jewel shop runs out of stock!" Shimmerbeam, blind altmer psijic: "Provided that I am confined to Artaeum, I do not think I will want for things to occupy my time." Shauna Blackfire, redguard necromancer: "Sounds like paradise. I hate people." Kirniel the Undying, cursed bosmer warrior: "I would feel useless, not being able to fight." Echoes-from-Dragons, argonian who thinks she's a dragon: "All the better to count my hoard!"