Still waiting for the next interview to be scheduled. Is it time to follow up?

Hey guys! In a bit of a pickle here. I've been interviewing with a job that I'm really excited about. It would be huge for me as I've felt stuck in my current job for a long time. I had my second phone interview last week and on Friday the recruiter told me they want me to come in for an on site interview and asked for my availability. I eagerly submitted my availability and now have been waiting for nearly a week for any sort of info regarding the on site.

Is it normal for some companies to take a long time to schedule an on site? Would emailing them asking about an update make me seem impatient? How would I word such an email? Thank you all for any advice! This has been stressing me out the past few days.

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