I have unfortunately been put in a position where my advisor has neglected to write me a letter of recommendation for a whole year. We have a good working relationship, and I have no reason to suspect that my advisor is unwilling to write a letter, however, for some reason or another it hasn't been done even after numerous requests (on my part) and assurances (on theirs) that it would get done. I was unable to secure a post doc in part due to this, and I am starting to become exasperated with how this situation is playing out. Has anyone faced a situation similar to this before? What steps would (did) you take to resolve this without burning any bridges? Does anyone have any advise on how to move past this in the event that I cannot get the letter?
I wouldn't say that you haven't been able to get a job due to 1 letter not being written. What about teachers? What about any internships? Is there anyone in the field that you know who can write you a letter of recommendation that has seen your work? Also for the record, letters of recommendation are becoming less popular due to the fact that usually they are written by people who will write it from a biased point of view and not objectively if it is for employment.
I always maintained a close relationship with all teachers so that, if one fell through or they were unable to, I had other teachers to go to. In terms of not burning the bridges, I think it would be best to be direct, without being offensive (as in being combative). Also it is possible that people are simply not comfortable with writing one but don't want to say no.
I would say something to the effect of:
We have a good working relationship. Something I greatly value and I respect your opinion. I understand for the last year, I have been asking for a letter of recommendation. I also understand that people are busy and writing one of these could take some time. I was wondering if you feel comfortable writing this for me? My opinion of you won't change if you actually decline to do so but if you can please let me know so I can see about getting it from other sources. There are a few jobs coming up that I would like to apply for and would greatly appreciate it if you can provide the letter or let me know if I should ask someone else.
you know? something that shows you value them and won't be offended, but also calls them to let you know honestly. Gut says, if they have been putting it off for a year, there is a good chance that they actually DON'T want to write it but don't know how to say no. The above paragraph gives them a door to kindly decline without hurting you and you don't burn any bridges. One thing I would avoid though is state how you lost out on jobs due to it. You don't want to create a negative aspect and it won't help him choose to write one or be more honest.
Some people simply don't write these letters. My programming professor that I had for 5 of my classes told me that he wouldn't write a letter. This was a bit shocking because he was someone that spoke very highly of me and allowed me to visit his other classes to help him with struggling students ( I was also the head tutor of the campus so I was able to legally TA anyways). He stated though that he didn't want to write these letters because they are hard for him to write. They put A LOT of stress on the writer to say the right words. If they say something wrong, they could end up being the reason you don't get the job. A lot of people don't want that kind of burden and will decline writing them. Many teachers want to see you get the job on your own merit and not based on someone else's words.
EDIT: OP has modified the question to reflect post doc. Though the thoughts about getting him to be able to comfortably state his honest opinion on whether he really will or not still should apply.