As a (soon to be) solo developer, I'm not an expert in marketing. Do companies exist who I could partner with to market my app, on a profit-sharing basis?
It's a pithy response, but marketing & sales are your business - at least 50% of it and possibly more.
Having said that, deals like this do get done. Sometimes with individual rain-maker sales people, sometimes with companies already operating in your niche who are willing to sell and market your product into their customer base.
I'm not aware of any companies that specialise in this. Most deals would be done on an ad-hoc basis.
My feeling is your product will need to be something really special with high barriers to entry for this to work. I would also expect to have your heart broken by the percentage you would have to pay.
I see affiliate marketing as a way to outsource marketing.
You set up an affiliate program, let the experts reach your target customers and give them a commission for every sale.
Don't do it! Echoing Benjamin, marketing IS your business. It's learnable like anything else. The best advice I can give is to study direct-response marketing. A good tactical marketing book would be one by Jay Abraham: http://www.amazon.com/Getting-Everything-You-Can-Youve/dp/0312204655. It's not necessarily written for the web per-se, but all of the principles apply in some way or another. Also, watch interviews with successful web entrepreneurs at Mixergy.com. You can learn a lot there.
Since you are selling developer skills then maybe the place to go is elance It may seem counter intuitive, but building a better mouse trap will not lead to the world bating a path to your door. Without marketing the world won't know what you are offering - let alone start looking for your door. In my experience (I have set up a couple of companies one of which pays the mortgage) start with a customer (there must be at least one of them) and work back from there. At least with elance you can find people who may value your skills without having to do any conventional marketing .