I thought that it couldn't demodulate DMR and other modes unless it had the hardware to do so. Feel free to correct me if I'm wrong, though.
You definitely can, but legality of such procedure is questionable. DMR uses AMBE codec, which require licensing from DVSI, but you can find open-source implementation on github (mdelib, jMBE). Projects such as DSD-FME[1], which allows you to decode much more than just DMR (P25, NXDN or SDRTrunk[2], which allows you to listen to trunking systems (p25, dmr tier 3 and much more) uses software implementation. OpenWebRX users software decoder too https://github.com/jketterl/digiham
Can also demodulate digital modes (DMR, YSF, etC) and it have nice, modern UI.
WebSDR have a weird list of requirements if you want to get install file - should open instance to public, provide lots of info, etc etc etc. It's not just closed-source.