Sure, most normal rivets start out as just short lengths of wire.
Almost nobody serious uses a rivet gun. (Some of the aluminum people
do use pop rivets for effect, but they’re the wrong scale and form
for serious wearable use.)
The one truly critical thing about using wire is that you have to
make sure that the hole you’re threading the wire through is very
(very, very) close to the exact size of the wire you’re using.
Nevermind if the wire’s supposed to be 16 gauge, measure the
actual piece of metal in your hand, get that diameter, and then go
get a drill bit to match it. (plus as few thousandths of an inch as
you can manage.) (like.005" or so. No fractions.) It really does
need to be tight, or you’ll have all sorts of problems.
This means you’ll have to get into what are called “number drills”.
They’re small drills that come in a variety of very small sizes,
that have very small steps between each different size.
What you probably want to do first is pick up a copy of Tim
McCreight’s “The Complete Metalsmith”. He covers wire rivets in
there pretty exhaustively. (Pages 101-105 in the current edition.)
The fact of asking this question tells me you don’t already have a
copy. Start with that. It’ll help. A lot.
There’s a reason it’s been the basic textbook for 20-30 years now.