Assuming you have the other materials to work with (stone, wood and glass), you can go on fine without metal. The most important use for metal is in armor and weapons. All other uses can be substituted - wood and glass for trap components, stone for mechanisms, glass and leather for vials / waterbags, etc.
The usual response to questions like this is "dig deeper!" but let's assume that there really is no metal on the map (which is fairly unlikely, but there's also the question of whether or not the following methods are more economical compared to a mining, smelting and forging industry). There are really two simple ways to get metal on maps without them.
You can buy weapon and armor directly from traders, but these are usually prohibitively expensive for an early fortress. Instead, purchase bars to forge your own weapon and armor. Cages are also good candidate for melting, according to this handy table at the DF Wiki. Remember to request for them when your outpost liaison arrives.
...is a cheap, renewable, environmentally friendly method for gathering silver, iron, copper, and more rarely other weapons grade metal. An early fortress should use a mix of cage and stone fall trap, and perhaps a pair of drawbridges to trap the goblins.
If you feel industrious, create wood or glass trap components (giant serrated discs and the like) to use in more efficient, reusable weapons trap. Older fortresses will sometimes have large, elaborate traps (example) using elements such as pressure plate, magma, water and ice to efficiently and safely mine out large quantities of goblinite. Remember that to extract every last ounce of goblinite, one should lure the entire squad of goblins all at once into the traps, because goblins have an unfortunate tendency to flee after a large percentage of their compatriots are annihilated.
The first few units of goblinite can be directly given to your fledgling military, or built into traps to allow more goblinite to be extracted. Later ones can be melted down and reforged into weapon and armor the silly goblins don't use, or as training for your metalsmithes.