Finishing Running Start (Dual Enrollment) with one prerequisite left?

I’m a junior in running start and I want to major in computer science at UW once I graduate from high school. By the time I graduate I will have an AA and all of the computer science prereqs done except for calculus 3 (math 126). I could get it done if I do a full quarter this summer, but that’s not really something I want to do (I want to get a job and maybe do some volunteering). So basically my question is, would not having this prereq done really matter, if I could just take calc 3 at UW if I get accepted? Also, since I’ll be applying as a freshman and not a transfer student, this shouldn’t hurt my chances of getting accepted, right?

As a frosh applicant, it is not expected that you will be ready to declare your major upon entry. You can complete remaining prerequisites after entering college.

Note that UW CS is extremely competitive for direct admission as a frosh. UW may also admit you as undeclared, in which case you face another competitive admission process (using college GPA and essays) to get into the CS major.

https://www.cs.washington.edu/prospective_students/undergrad/admissions
https://www.engr.washington.edu/current/admissions/admitstats

No, it will not hurt your chances of acceptance at all.
Most Running Start students do not finish an associate degree, so you have shown a lot of academic rigor by getting that done.
That being said, all the CS direct admits I’m aware of have strong math transcripts, so if there is one class you have to be short on, Calc 3 may not be the very best choice.

But trying to finish high school requirements plus a full transfer program is hard and it is very common for RS students to not be able to do it all. They definitely still get in to UW, and you will be in strong shape for direct admit for CS if your grades are good.