You would be the one to apply for the work permit (not your employer). Your employer needs to give you a job offer and generally also obtain an approved LMIA before you can apply for a work permit. Yes, you can apply for the work permit while the visitor extension is pending.
You should leave the visitor extension processing since this is what is keeping your status valid in Canada (implied status) and you need to have valid status to apply for the work permit.
Talk to your potential employer as they most likely need a LMIA. They need to prove no Canadian or PR could be found for the job and pay $1000 for the application fee. This takes many months and then if successful, you apply for a closed work permit. Again many months of processing.
Seems like you are already a PR so this question maybe for a family member. They should be aware that finding an employer willing to go through the LMIA process and hire needing a WP is very tough. Wait times to receive answer on WPs are many months long unless you have very specialized skills that are in demand.