I would think that human/animal hybrids living in the sewers would have difficulty getting hold of swords or a bo staff or the materials to make them- was this ever addressed in any of the comics or tv series?
I vaguely recall the recent movie saying that the turtles found their weapons in the sewers, but I could be wrong.
However, in a city like New York, you could probably find low quality versions of each of these weapons by searching through garbage for a relatively short time. They certainly wouldn't steal them, at least in the current cartoon series (which began in 2012).
In the 2012-present series, they seem to lose weapons pretty frequently. In the original movie, on the other hand, Raphael's reaction to losing a sai suggests that he doesn't expect to get an opportunity to obtain another one anytime soon.
At the very least, it would be fairly easy to find enough money in the sewers to buy the weapons.
I saw from the Nick series episode Lone Rat and Cubs that the Turtles got their weapons because of their actions with them:
- Leonardo was holding onto Splinter's katana as he was pushing them in a cart in the sewers, running from the Krang.
- Raphael got his sais after he fell from the cart because a sai was lodged in a wheel, and he picked it up off the ground.
- Donatello got his Bo staff because he... well, the episode didn't really say why he got his Bo, because there was one in the cart, but he didn't interact with it.
- The youngest, Michelangelo, got his nunchucks because when Splinter reached for his chucks to fight a Krang droid, it looked at the end, where Mikey was sucking on it. One of the other million things I found cute in the episode.