I just did one for a missions organization a few weeks ago. Sometimes they are really on the ball and you get confirmation within a week here. One thing that Techsoup looks for is having a website up on the same domain, Techsoup will glance sideways when you don’t have a website or the website is using a different domain than the email (I know, it’s weird, but there are some organizations out there that insist on doing that). Definitely don’t use anything to contact Techsoup other than an email address using their registered domain. Granted, this was APAC Techsoup and the organization has a Singapore nonprofit status so it will be interesting to see how long it takes in the USA.
And yes, you can still start off with a trial within the tenant and you can cycle through all the various M365 and O365 products if it really did run into a snag.
UPDATE: Last week (November 2019), an IT service provider I know in the USA made application on behalf of a synagogue and it only took 2 days to get verified. It took longer to convince the Rabbi that they were eligible for donations than it took to get verified for the donations. I’m pretty sure that anybody that goes longer than 4 weeks is an outlier rather than the norm.