Is there a way to dynamically pull company logos for banks and brokerages into a webpage without individually downloading each logo or building a list of URL's to link to them directly? I'm looking for any site that has an API to get company logos given a ticker symbol, company name, or website URL.
No, there isn't one.
You could build it, but note that most companies have restrictions on the use of their logos, so you'd need to have an agreement with each of them (that is probably why no such service exists).