Browse data on old machine's HDD, without booting machine or removing HDD? For my current laptop I have a 2017 MBP running macOS 10.14.

I have 2 older Mac laptops which I wish to get data off, one of them is a 2007 Macbook and the other a 2004 iBook.

Other than disassembling them removing the HDDs and putting them into enclosures, is there a way I can browse the data on these older laptops from my current laptop? Perhaps by connecting my current laptop to the old laptop via USB? If so would the old laptops need to be powered on?
You can boot each older laptop into Target disk mode by holding down the T key at startup. The older laptops need to be powered up. This will present the internal HDs on these older laptops as an external HD on your MBP.

Both of these older laptops have FireWire 1 ports, however, so a physical connection to your current laptop is problematic as you will have to connect a FireWire 1 cable to a FireWire 1 > FireWire 2 adaptor, then connect that to a FireWire 2 > Thunderbolt adaptor to your MBP.

This may or may not work, but I used FW1 > FW2 adaptors for a few years without a problem.

If either or both of the older laptops do not power up, then removing the HDDs from within them will be necessary.