You're definitely in a bit of a bind here as you can't really install new devices (the cards) without loading the drivers...and to load the drivers you need a floppy or a CD. You may be able to use a PCMCIA hard drive, but that's iffy. Since it's a 486 with '95 all the possibilities of a USB solution are unfortunately out.
I assume the laptop does not have a network card installed. Does it have a modem? If that's the case you can try dial up and downloading needed drivers. If so, the next thing I'd get is an external CD-Rom drive. Download the drivers, install the drive, and then use the CD to load what you need. There are PCMCIA drives, scsi, and parallel (yuck) all of which you could use.
If you don't have a modem installed, check and see if the previous owner put the .cab files on the machine. If so, you can connect devices up that have drivers built into the OS. This will probably allow you some older basic modems and network cards. Try to pick one of these devices up on Ebay (should be dirt cheap) or try using a generic driver with whatever you have access to. If you can configure the card/modem, you can then use it to connect and get the drivers you really need.
If you don't have the .cab files, you don't have a modem, you don't have a nic, floppy or CD, you're getting into real muddy water here. You *could* get a 2.5 hd adapter, remove the hd, connect it to a desktop, configure the drive there, put the drivers, cabs etc on it then move it back to your laptop*cringe*....I'm not saying I would...just that you *could*.
This'll be a hassle...maybe more than a 486 laptop is worth.
Anyhow...good luck. Hope this helps.