Simple experimental design of a chemostat without pumps
Here is a link to a design in which electrolysis is used to generate gas which displaces liquid from the reservoir into the culture chamber, which in turn displaces culture into an outflow (where samples could be collected). The design does however require a pump for aeration, and I can't think of a way in which this could be avoided, although there is a reference cited in this article which apparently describes the use of gravity flow (I haven't checked it out).

Just for interest, this link is to a design which does require a peristaltic pump and a vacuum line but it looks to be achievable.