Not much they can do if they are on a shared server and the resources are running low for the host server. Then there are dozens of potential factors which may slow down the response of a server. slow application logic, slow database queries, slow routing, frameworks, libraries, resource CPU starvation or memory starvation.
If on a shared server it could be another client on the server with a malformed configuration of their software that's causing a high impact on server resources, over selling of server resources will cause problem, meaning too many clients on one server using up server resources. A SQL server data base can only process so many query at a time and when that limit is reached the database puts the query in memory with a time stamp and executes the request in order. Running a trace-route to see where the data bottleneck is will tell you where the slow down is.
I see they are running their domain through Cloudflare and I have had problems with slowdowns of late running domains through Cloudflare.