Yes, they should get permission. A company logo is protected by trademarks and copyrights in
almost all jursdictions. Companies do not like to see their logos splashed across their vendors' websites, case studies, presentations etc., without their legal/communications teams having a say over that. Most Fortune 500 companies would have this as a standard legal clause in their agreements and require that such requests be made in writing.
(This excludes public brand usage guidelines published by companies for including their logos eg: Social media links for FB, Goog, LinkedIn, Instagram etc., and they are covered by the separate terms of use there.)
Check your agreement with your client. If it has such a clause related to marketing material (social proof would fall under that), seek their permission as prescribed in your agreement. It may help to get your client liaison on your side by letting them know first.
Then wait.
It is quite possible that the client may not get back to you. In that case, err on the side of caution and DO NOT use the logo on your website/social proof page.
As an alternative, when we made presentations to a small room (of prospective clients, potential investors etc.,) we may include a Top clients roster slide with the logos but mark that specific slide as confidential and show it only within the room and never use it for broader distribution. This practice is easier for your legal team to clear if the parties in the room are under non-disclosure agreements. It may be okay in other circumstances; read the room and use your discretion.
Not a lawyer.
Source: personal experience.