My suggestion:
Find a network card with a PLCC socket, install in your PC
Find the software you need to enable the socket and/or define where it appears in the memory map. (Probably requires a boot into MSDOS).
Install the device in the network card socket, reinstall the card.
Boot the PC into MS-DOS with some form of writeable storage (floppy drive, small hard drive with FAT16 2.1Gb partition).
Use the instructions at
http://mess.redump.net/dumping/dump_bios_using_debug to find and dump your ROM image.
Note that a 29EE00 is a 128k x 8 device, and some network cards may not support 128k boot roms.
The other alternative is to look for an EPROM reader - either use an old PC with a parallel port and something like the EPROMr (without the Vpp generation), or search for an Arduino based solution.
Note that I haven't done this myself - I have an EPROM programmer, and a PLCC32 to DIP adaptor.
If I didn't have that available, and the "put the device in a network card failed", I'd build myself a reader using an Arduino with a couple of counters to generate the address bits. (something like:
https://www.element14.com/community/blogs/SalsCorner/2015/07/27/eprom-reader-project )
PS. I assume you've got the device datasheet at
https://www.eit.lth.se/fileadmin/eit/courses/edi021/datablad/Memory/Eeprom/29ee010.pdf