In my first two roles I found that more senior level developers gravitated towards mentoring me anyway. I didn't really even have to seek it out, usually just getting in the door and humbly admitting to yourself that you don't know as much as you thought you did is enough for people to want to share their knowledge with you.
Don't be afraid to ask questions in the workplace when you're new, that's the only cardinal mistake you can make, the mentorship will come automatically.
Some things that might help:
- ask lots of questions, listen, and take notes - be humble and gracious when you receive feedback. Most people avoid mentorship because its really hard to deliver negative feedback delicately. Easy to avoid it altogether. Ask for feedback actively, and start with: "what can I do better here?" or even better, before starting "how would YOU attack this problem?" - scout the organizations you want to work for ahead of time. Creep linkedin profiles, etc. Do you have similar interests to the senior devs (which will allow you to connect on personal level)?
Professional institutions typically do too.
Either option takes the burden off asking someone, since you're merely applying to a pool of people that have declared a priori their willingness to be a mentor!