I could very well have missed something obvious, but from what I've gathered from watching the Railgun anime, I can't recall any clear indication that Touma knows about why Misaka gave her DNA to Academy City researchers- that she was lied to and thought it was to help children with muscular dystrophy (ep2 at 0:00). If anything, what I've observed indicates that he would be more likely to think that Misaka gave her DNA to those researchers while being fully aware of their true intents.
At the end of the Sisters arc, he seems to have come to a positive mental outlook on the happenings, which he passes on to Misaka to ease her conscience, without further questioning.
It feels off to me. From what I know of Touma, he often confronts people's ideas and intentions, questioning their thoughts and (usually) finally punching them in the face to "destroy their screwed up delusions". Ex. Misaka in ep 14 at 13:25 (no punching here, thankfully), and Accelerator in ep 15 at 14:00.
Here's the timeline of what I'm aware of Touma learning about the origin of the Sisters:
ep 13 at 19:05 (Touma's first encounter with the many Sisters after seeing the dead body of Misaka 10031)
Touma: Who are you?!
The Sisters (10032 speaking): Cellular clones created as mass-produced military models of our sister, Misaka Mikoto, one of only seven Level 5s in Academy City, known as the Sisters, answers Misaka.
ep 14 at 7:07 (Touma reading the paper he found hidden in Misaka's bear stuffy)
Radio Noise... How to Shift Accelerator to Level 6... Level 6? What's that? Out of Academy City's seven Level 5s, only Accelerator is capable of hitting Level 6. As a result of the Tree Diagram calculations, 128 different battefield types will be prepared, and by killing Railgun 128 times, Level 6 can be attained. Railgun? Isn't that Biribiri? Naturally, 128 Railguns cannot be prepared. As such, mass-produced Sisters were developed. Therefore, when Accelerator kills 20,000 Sisters, he will achieve Level 6...?
Misaka herself learned that the original Radio Noise project failed and was passed on to the Level 5 Shift researchers in ep 5 at 14:38.
ep 14 at 11:26 (The bridge scene with Touma and Misaka):
Misaka: But if you've got that, (referring to the Level 6 Shift paper, which Touma brought with him to the bridge) it means you went to my room without asking me, right?
(Misaka's Internal dialogue, as if speaking to Touma and herself): The way you see it, I'm a participant in the experiment, who volunteered her DNA. So have you come here to condemn me? [...] Oh, well. Even if the facts are a little off, the end result is still the same. That actually makes being blamed by someone else a whole lot easier.
Misaka: So, in the end, you think what I've done is inexcusable, huh?
While the full contents of that Level 6 Shift paper were never shown, Misaka's thoughts here indicate that the reader of the paper would be led to disdain for her actions instead of sympathy. Disdain would make sense if the paper indicated that she knowingly volunteered her DNA for the Level 6 Shift project, or if the paper went further back and gave the same history on the project as is shown in the wiped data Misaka found in ep3 at 17:30- that the original intention with her DNA was to "resolve a genetic sequence pattern that gives rise to a Level 5, allowing a Level 5 that is born by chance to then be produced with 100 percent certainty."), or if it was left vague such that one might suspect that instead of what she was told by the Radio Noise researchers: that it was to help children with muscular dystrophy, which would naturally give rise to sympathy, because she was lied to and taken advantage of for evil purposes.
Though Touma responds with "What are you on about? I was concerned, of course.", in context, and taking into account his caring and sympathetic character, that doesn't seem to indicate to me that he somehow knows the real reason why Misaka gave her DNA to the Radio Noise researchers.
In ep16 at 12:20 (Misaka visits Touma in the hospital):
Misaka: But still, because of me... So many...
Touma: If you hadn't offered your DNA, none of them would have been born. The experiment was wrong in a lot of ways, but you should at least be able to take pride in the fact that they were born. If they hadn't been, they could never have felt happiness of sadness to begin with. Besides, they all wouldn't want to see you moping around all by yourself.
He seems to come to a happy ending by an entirely different thought process, which is fine and dandy, but if he doesn't know why Misaka originally gave her DNA to the Radio Noise researchers, I'm quite surprised that he seems so unfazed/forgiving and doesn't question deeper about why she gave her DNA- again, given that the Sisters were designed to for Academy City's military purposes, and then to level up Accelerator, and that such a big part of his character is to confront the intentions of those to act villainously, or who support or allow evil to happen.
To recap, from what's shown in the Railgun anime, Touma knows that the Sisters are military clones of Misaka. He knows about the Level 6 Shift project, and there's a possibility that he knows about the Radio Noise project (but not necessarily and not likely how exactly those researchers got their hands on Misaka's DNA. I doubt a researcher would find it important to admit in a research paper that they lied to a child to get their DNA).
Does Touma know about why Misaka gave her DNA to Academy City researchers?
I haven't read the light novels or the manga adaptations, so there may be information there that I'm missing. I've watched later parts of the Index anime, but not the first season (not any parts that overlap with the Sisters arc of Railgun).