@Andy Kay :- Hallo.......and 
to the "kennels"!
If what @wizard and @cobaka have discovered about your machine is true, then I'll eat my hat if it won't boot from USB. Truly I will!
I had an even older Dell Inspiron from around 2002, with a single-core, 32-bit only Pentium 4.....and that would boot from USB. Mind you, I will admit.....Dell always were early adopters of new technology, and that Inspiron was one of the very first consumer laptops on the market to come with the USB boot feature. Some manufacturers didn't get around to incorporating this ability for another 3-4 years after that.
By the time the Core2Duos came along, their chipset supported the feature as standard.......yet even the previous-gen Pentium Dual-Cores would perform the same "party piece", so this shouldn't be a limiting factor in any way. 
M'colleague's points about the FDD might be worth bearing in mind. Even if that doesn't 'play ball', there's still the PLOP option.......or, you could splash out a wee bit of cash on a cheapo SSD. Doesn't need to be owt special; so long as it'll fit inside, connect, and the BIOS/OS 'sees' it, you're in business.
There's a whole RAFT of options available yet.....
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It's probably fair to say that most of the forum "regulars" here all have 'our Pup' installed to & running from an internal drive of one sort or another. But with Puppy, this is NOT a hard'n'fast requirement, by any means. In fact, you'd be amazed at some of the weird & wonderful hardware combos we DO like playing around with.......often, stuff that any "normal" Windoze user would have tossed in the trash years previously! Just because it won't run Windoze, does NOT make it useless....
Puppy can run on almost anything. Literally. Your machine is a prime Puppy candidate; all you need is a drive of some sort.....whether external flash/HDD/SSD, makes no difference ATM. A USB flash drive will be the simplest and easiest route, I think. Despite having USB 2.0 ports, I'd recommend a USB 3.0 flash drive; they may not "match", per se, but a USB 3.0 in a USB 2.0 port will still run faster than a USB 2.0 will. Experience has taught me that!
(Here's an example:- The anciente Inspiron finally died a couple of years back, but for the last 5 years of its life I'd been running an SSD in it, designed to work with the older IDE/PATA standard. When the Dell made its way to the great scrapheap in the sky, I rescued the SSD before she disappeared.
That SSD is now sitting in an enclosure I made out of an old Compaq floppy disk storage box I'd had sitting around for ages. Courtesy of a PATA-to-SATA adapter - £2 off eBay - it connects to my big HP desktop rig via a SATA-to-USB 3.0 adapter cable.......and is currently booting ChromeOS (just for the hell of it)! 
)
"Our Pup" is nowt if not versatile!!
Mike. 
