Select a sample size form the infinite population

There are two issues here, (1) Getting a confidence interval of the desired length and (2) getting a representative (unbiased) sample of the population.

For (1) it is often enough to use a formula to find the $n$ that will result in a confidence of the length you want.

For (2) the task is more difficult. If your list is comprehensive, then a simple random sample from it may be all you need to estimate the proportion of such chains that have at least 200 stores across the country. That is, $P(X ge 200,|,X ge 100),$ where $X$ is the number of stores a chain has.

However, a random sample from your list will probably not give an unbiased estimate of the proportion of large $(X ge 100)$ chains that have a store in a specific location, such as Sikeston, MO. A small chain $(X < 100)$ serving mostly the south and midwest of the US may be much more likely to serve a particular small town in MO than will a huge chain with most of its locations concentrated in the contiguous 11 western US states. But a few very large chains have locations almost everywhere (e.g, McDonald's, Walmart, certain drugstores, and truck rental companies, etc.)