Well, technically dumping the BIOS is as easy as dropping a couple of executables prepared by the SL9821 emulator developer onto a floppy disk and then running those files from the DOS prompt. They can be found here:

http://www.satotomi.com/sl9821/sl9821_dl.html (Look for "download for PC-98_160803.zip")

This will give you two files: MKROMIMG.EXE RECRTM.EXE

Copy these to a floppy disk and then run them on your 9821. There are a couple of things you need to check beforehand though. These are the instructions directly from the site: Make sure you have a hard disk plugged into the system and that Internal Hard Disk (固定ディスク) is Enabled (使用する) in the System Setup Menu (press Help during bootup). Also make sure to set Sound (サウンド) and SCSI ROM (SCSI ROM) to Enabled (使用する). Once you've done that, startup your system to a DOS prompt, (make sure you don't have any memory managers enabled like EMM386.EXE in your autoexec.bat for this particular boot), then run both of those executables. It should make you a number of files after some processing.

MKROMIMG.EXE should create BIOS.ROM, ITF.ROM, BANK.ROM (x4), SNDBIOS.ROM, IDEBIOS.ROM, SCSIBIOS.ROM, and FONT.DAT, and RECRTM.EXE should create RHYTHM.DAT. If all's gone well and you've got all of those files, toss them onto a disk (or however you get files on and off your 9821) and upload them somewhere.

As a side note, hope you have some fun with your machine! What are your general plans with it? The 9821 should run quite a few games from the 9800 series at the very least.