Indeed, this is something which needs to be done on the server side (by your hosting provider, if you don't self-host). PHP has some modules which are compiled into the executable and a number of dynamic modules (extensions) which may be loaded on startup (specified in the configuration file). The mbstring extension is one such module. Without it loaded, any applications which depend upon it will fail.
I'll add a note about this dependency in the docs. As mbstring is a pretty "standard" extension (i.e., shipped in the typical PHP distribution and used by many things), I hadn't even considered that it might not be loaded in a standard web server configuration (not that I was specifically aware of Simple HTML Dom's dependency, though it makes sense; anything doing string matches with potentially multi-byte character sets is likely going to want it).
Again, good luck with this.