Hi John! I wouldn't ask if I could take the position. Sounds quite abrupt. I'd say something like... "I'm interested in applying for a position in your company." You could even ask the recruiter how you would go about applying for the position or expressing your interest.

Hello Evan Mildenberger! Thank you so much for your comment! Good luck!

Your sentences are grammatically correct. But as someone has mentioned, we are not very direct when we are trying to seem polite. So "Can I apply for / I would like to apply for this position" are more polite since it doesn't appear like you believe you expect to get the position.