I built flashrom from commit 7b8b945a004ba69529fe0283ad18a40ee34255bd

I was able to read firmware from my pci card. 

I don't have an original firmware (version 013600) to compare it to but it looks legit:

cmp -l vl805_fw_013600.bin vl805_fw_013701.bin | head
  130  40  41
  131 256 245
  133  42  43
  134 340 147
  151 214 374
  172  74 303
  178 166 346
  180  43  44
  181 202  11
  241  42  41

flashrom dumps the whole 512KB of FM25Q04 using generic SPI. 

Comparing other official firmware my guess is the layout(?) for this firmware should be:
0:17AD7 FW013600




flashrom unknown on Linux 4.15.0-130-generic (x86_64)
flashrom was built with libpci 3.5.2, GCC 7.5.0, little endian
Command line (7 args): ./flashrom -VV --programmer vl805 -r test.img -o test.log
Using clock_gettime for delay loops (clk_id: 1, resolution: 1ns).
Initializing vl805 programmer
Found "VIA VL805" (1106:3483, BDF 01:00.0).

...

SFDP has autodetected a flash chip which is not natively supported by flashrom yet.
All standard operations (read, verify, erase and write) should work, but to support all possible features we need to add them manually.
You can help us by mailing us the output of the following command to flashrom@flashrom.org:
'flashrom -VV [plus the -p/--programmer parameter]'
Thanks for your help!
===
Found Unknown flash chip "SFDP-capable chip" (512 kB, SPI) on vl805.
Probing for AMIC unknown AMIC SPI chip, 0 kB: compare_id: id1 0xa1, id2 0x4013
Probing for Atmel unknown Atmel SPI chip, 0 kB: RDID byte 0 parity violation. compare_id: id1 0x00, id2 0x00
Probing for Eon unknown Eon SPI chip, 0 kB: RDID byte 0 parity violation. compare_id: id1 0x00, id2 0x00
Probing for Macronix unknown Macronix SPI chip, 0 kB: RDID byte 0 parity violation. compare_id: id1 0x00, id2 0x00
Probing for PMC unknown PMC SPI chip, 0 kB: RDID byte 0 parity violation. compare_id: id1 0x00, id2 0x00
Probing for SST unknown SST SPI chip, 0 kB: RDID byte 0 parity violation. compare_id: id1 0x00, id2 0x00
Probing for ST unknown ST SPI chip, 0 kB: RDID byte 0 parity violation. compare_id: id1 0x00, id2 0x00
Probing for Sanyo unknown Sanyo SPI chip, 0 kB: RDID byte 0 parity violation. compare_id: id1 0x00, id2 0x00
Probing for Winbond unknown Winbond (ex Nexcom) SPI chip, 0 kB: RDID byte 0 parity violation. compare_id: id1 0x00, id2 0x00
Probing for Generic Variable Size SPI chip, 64 kB: Probing for Generic unknown SPI chip (RDID), 0 kB: RDID byte 0 parity violation. compare_id: id1 0x00, id2 0x00
Probing for Generic unknown SPI chip (REMS), 0 kB: compare_id: id1 0xa1, id2 0x12
Found Unknown flash chip "SFDP-capable chip" (512 kB, SPI).
Block protection is disabled.
Reading flash... done.