http://www.dmlp.org/legal-guide/using-trademarks-others
"Although the practice is widespread, it is beginning to draw fire from trademark owners. As noted above, late last year a representative of Avis requested that law blogger Eric Turkewitz cease-and-desist from using its logo in a post about a lawsuit involving the company (see Turkewitz's post). The law is not entirely clear on this point. The nominative fair use defense may not apply because using the logo is not strictly necessary for describing the trademark owner or its products or services. Nevertheless, courts would probably find that use of a logo in the process of news reporting is not likely to confuse consumers."
In the context you describe, there could be more potential confusion than in news reporting, since there may be doubt about whether using a company's logo on your site implies some kind of endorsement by that company (even if you don't explicitly claim that it does).
The easiest thing, of course, is not to use those companies' logos (just refer to the company names in text). If you want to do use the logos, you'd be much safer if you either got permission or consulted a lawyer.
IANAL. I am someone who has a history of being a stickler for asking permission first, rather than forgiveness later. There are some downsides to that. It can be stultifying and sometimes is overly cautious for purposes of getting things done.
So, some thoughts:
Big companies may view such things as free advertising. But I would guess whether they are okay with it or not would depend a lot on whether or not they like what you are doing, how careful you are to clearly signal that you are merely using their public API and otherwise have no relationship to them, etc. If you are trying to ride their coattails and imply that they endorse your work when they don't, that might not end well, especially if they don't like the kind of work you do.
Likability of your work can hinge in part on framing. For example, the same service might be framed as protecting the privacy of vulnerable people or it might be framed as enabling criminal activity in defiance of a government you don't like. So one thing you might want to do is think very carefully about how your own materials talk about your work and making sure to replace phrasing with unfortunate connotations with different words that lack the unfortunate connotations.
If the companies complain, you can promptly take it down. Assuming you haven't done anything nefarious, that should be sufficient.
Companies often have brand usage guidelines that you can find online. Try searching for those first. I'm sure you're not the only person who has wanted to show that your service runs on AWS.