Really, there's two sane ways to do it: you can either bring an anvil and plan on forging an axe out of raw materials, or you can forget the anvil and buy one from the first caravan. Odds are good that you won't get a fey mood for a metalsmith the first year (though to be safe, don't make any of your starting Dwarves skilled in any metalworking professions), and you can make 1,500 (or 4,500 if they only bring steel) dorfbux worth of goods for trade easily in 6 game months' time.
Here's what I tend to do: I drop the anvil and boost my other resources. Once I've arrived and set up base, I have one Dwarf focus on making mechanisms (they sell for a disproportionately huge amount, especially when made from obsidian or flux material). The rest do other preliminary setup of the fort, and by the time the first caravan has arrived, I've at least two dozen high-quality mechanisms for trade and the anvil is easily gotten (as well as anything else I need, but that's the beauty of a no-anvil embark: you can take along more of other supplies and avoid needing to buy too much from the first caravan as a result).
If you don't have flux stone or obsidian handy, that makes this a bit trickier, but you can manage even then. A stone carver can churn out hundreds of stone mugs, which don't sell for much individually, but do net a big profit overall. Just be sure you have a lot of free wood to build the massive number of bins that tactic requires, too.