What is a good way of getting peer feedback on my draft cover letter? I am preparing material for the academic job market and I would like to receive feedback from peers on my cover letter.

Additionally to the traditional personal contacts, I have thought about online forums as a possible location, but have not been able to find any that seem good.  I am also not interested in websites that offer consultation from professionals in return for money.

How would you advise I go about getting such feedback?

Unfortunately, not everybody works in big departments or in universities with staff of the human resource management trained for this purpose. Also, it is the end of June: in some countries (in my case Sweden) this means the vacation period is started and it is taken very seriously (vacation=no working activity of any type, besides for doctorate students).
The most German universities have career service departments for questions like this. If this is uncommon in Sweden, I agree with Kimballs comment: show it to people you trust.

In addition: the Internet is full of resources like guides and tips. Just be aware of possible cultural differences that may appear between countries. A guide targeting to US audience could suggest or advice elements that could count as "no gos" in other cultures. If you want to check your cover letter against some online guide I suggest you to search one in Swedish that considers the national or cultural specialties.