I'm not aware of any web-based interface that will allow you to assemble your code... and execute it, I suppose... Do I understand what you're looking for?
If the boss doesn't want you downloading Nasm... well, it's the boss' computer. Can you get a large email attachment? I could arrange to mail Nasm to ya - tell me what build you want - fbkotler at verizon dot net. I suppose I could assemble your code for ya, but probably not produce an executable (except dos .com files or Linux), if that'd help any.
You really need to get your own system... and connection. A "junk" machine ought to be fine for experimenting with asm - don't need the latest hardware...
Best,
Frank