A few great points on this thread, but in some cases the context was missing (i.e. size of company, timing, objective, etc.), I'll try to add it below.
My assumptions based on the info in the thread:
#1 you are an early stage startup (i.e. still a young business with an initial customer base)
#2 you are looking to optimize your acquisition/growth efforts (i.e. increase pipeline of potential customers and improve conversion of those already in the funnel)
#3 you focus on the high ed space (where like with other large orgs, getting an approval can take ages and is less likely given your size and type of product)
#4 purchasing (or using) your product doesn't require a complex decision process or a significant budget.
Given #1 and #2, the use of logos (as well as names, testimonials, seals, certifications, etc.) is going to be important from a marketing and conversion stand point. such validation points play a key role in getting consumers over the line and can significantly improve landing page conversion/optimization. there are some differences in the psychology of b2b vs b2c customers when it comes to conversions but at the end of the day all of the decisions are made by humans and we all convert better given an offer that resonates with our needs, mitigates our risk and plays on our biases.
So the interim conclusion is that you need to have the logos out there... in fact, you will also need them in your decks (partners, investors), marketing materials and any other place where validation is required to improve the % of someone voting in your favor...
For a different product/business (e.g. heavy enterprise software, small/closed ecosystem) or at a later stage of the company (e.g. well known brand) the use of validation points can be less critical and may be more limited.
So now it's a question of asking for permission or not.... to me, this is mostly a matter of balancing the risk (not high) and resources (limited).
As a small startup that needs to execute fast and has limited resources + given bullet #3, there is no point in getting approvals from each university. that's why most similar startups in this situation prefer to simply do it and if anything goes wrong, ask for forgiveness. from a practical stand point, based on bullets #3 and #4 the risk you are taking is not too significant, it's not likely that one of the institutes will take you to court for using their logo and if they reach out (i.e. friendly email or C&D letter), you can remove the logo, apologize quickly with a look of surprise and defer to one of the "defenses" below.
If you want to play it a bit more safe, you can add a few "defenses" that can allow you some plausible deniability in case anyone asks (not from a legal stand point, more from a moral and best practices stand point):
* get testimonials/blurbs from your consumers (=employees) - even short ones.... in many cases the admins will be happy to provide a testimonial where you can use their name/title and they won't even know if they need to ask for permission from their employer... (your product is not expensive and used as a daily tool, it's not something they will need to think too much about). this will allow you to use the name of their employer as part of that blurb and a justification to add the logo in that context. you will be surprised how easy it is, most consumers would be glad to help if they like the product.
* use some text that distinguishes the specific use of your product next to the logos, e.g. "as used by employees from the following institutes" vs "as used by the following institutes". do not be blunt and say that employer X is directly using your service if this is not true.
* since your main focus here is better conversion on the landing pages, use smart design that would put the right emphasis on the call to action areas and have the "defense" elements in a visible but less prominent location on the page.
The suggestion to add some publicity language to your license/TOS that will allow you to use the marks is a good one in general but keep in mind it's mostly a way to cover yourself with small print in case someone big finds out and gets mad at you for using the logo. in reality most of your costumers will not read the TOS + if the employer doesn't like the fact the logo is there, there is no TOS document in the world that can help you keep it on the site...
As the company grows you will add more validation points and new marketing materials, including case studies, PRs, mutual promotions, etc. producing them will take some more time and you should use them to get the employers more involved and provide you with direct endorsements (instead of hacked ones), these will come with publicity clauses and proper marketing support. but until then, you can hack...
Good luck, happy to provide more specific advice if needed.
Do not use any of the above as legal advice, just a few insights based on experience in similar situations.