This answer is a Wiki! Edit it to add more suggestions.
The WILDLABS acoustic monitoring group is a good place to look! Lots of job & funding opportunities posted.
The Conservation Tech Directory is also a good place to look for both job boards themselves as well as organizations, companies, academic labs, etc. that do applied bioacoustics or passive acoustic monitoring - you can search by keywords & filter by different parameters.
The bioacoustics list serv ([email protected]) is a good resource for job postings. Instructions to join can be found at the list servs FAQ here but those instructions seem out of date (from feedback here) so you can contact the Holger Klinck at the K. Lisa Yang Center for Conservation Bioacoustics via email to be added.
The International Bioacoustics Conference (IBAC) has a listserv as well, similar to Cornell's listserv that Selene referenced above! You can subscribe to it through the website (look on the right-hand side for the link to join the listserv).
If you subscribe to MARMAM Digest, they send out job listings (international) as well as other marine mammal science related information: https://lists.uvic.ca/mailman/listinfo/marmam.
There are companies that hire bioacousticians and do research. JASCO Applied Sciences provides consulting and research services for assessing and mitigating underwater noise: https://www.jasco.com/careers. SMRU Consulting may also hire people with bioacoustics backgrounds: https://www.smruconsulting.com/.
Aquariums and ocean institutes also often have acousticians doing research.
When at conferences and workshops (such as the upcoming Biennial Conference on the Biology of Marine Mammals) it can be helpful to take note of the various organizations people work for, e. I just noticed someone was fromg., Ocean Science Analytics (https://www.oceanscienceanalytics.com/our-services), which I had never heard of before.
Mostly for UK but check in on the UK Acoustic Network (https://acoustics.ac.uk/ ) - Bioacoustics Special Interest Group (SIGBio) - for job postings updating regularly: https://acoustics.ac.uk/sigs/bioacoustics-sigbio/. PS: free for anyone to join.