Short answer: yes, if you control a web server under that domain name, and Let's Encrypt external bot can access the web server, or if you control DNS records for that domain.
The DNS validation may be tough (you probably do not control the DNS server, and it may be administratively challenging to obtain the necessary DNS records). So your other option is to use HTTP validation. Citing Let's Encrypt "How It Works" article:
These are different ways that the agent can prove control of the
domain. For example, the CA might give the agent a choice of either:
Provisioning a DNS record under example.com, or Provisioning an HTTP
resource under a well-known URI on https://example.com/
If you need to do so for an Intranet-only domain, you can setup a temporary/dummy web server for certificate generation purposes, and bring it up only when renewing the certificate. I suggest just buying a public domain and getting a certificate for that domain, and pointing it to the internal IP address. Please note that doing so may expose internal IP addresses, and you should do that only after a risk assessment and after getting an approval from the university.