First of all, it's important to note that the SSR didn't find Bucky's body. The Soviets found him, and gave him to HYDRA initially. Even though Zola was working for SHIELD using SHIELD resources doesn't mean SHIELD "gave him" Bucky's body to fix.
Secondly, at the time that Armin Zola was working on Bucky, he was not considered an "enemy". SHIELD probably genuinely believed he was working for them the whole time. Plus, the fact that HYDRA had already begun infiltrating SHIELD, means Zola likely had support from higher-ups to keep suspicion off of him.
It may sound a bit crazy to us, but what happened with Zola is very similar to what happened in real life at the end of WWII. Lots of scientists who were originally working with the Nazis in Germany "defected" to the US.
At the time, German scientists (especially physicists) were some of the most advanced and skilled in the world at the time, and the US really wanted their expertise -- and more importantly, they didn't want the Soviets to get them. Operation Paperclip was a very real military project designed to recruit as many foreign scientists as possible (I believe the number was well over 1500). In the vast majority of cases, these scientists were not particularly zealous supporters of their previous regime: they either worked for Hitler out of fear, our of desperation, or because it was where the really interesting work was happening.
With the MCU, this is basically how Zola came to work for SHIELD. After he was initially captured, Zola exhibited none of the usually traits of the HYDRA operatives the SSR had dealt with: he didn't kill himself, he gave up information freely, and he even rolled over on the Red Skull with little prompting. He then served a few years in prison, during which he was presumably a model prisoner.
When SHIELD started up, they would have needed scientists with experience dealing in the kind of unusual stuff that SHIELD deals with, and Zola was right there. Zola kept up the appearance of being a fully reformed SHIELD agent, much the way the other HYDRA moles did. SO it would make perfect sense that SHIELD would want Zola to come work for them, and give him access to the resources he needed to work on Bucky.