We should distinguish between the "big, interpretive argument" — why were they bombed? — and the kind of "piece of evidence" that you would find in a primary source. A big argument is an interpretation of many primary sources. Any argument worth arguing about has a raft of them. So if you read Gar Alperovitz's The Decision to Use the Atomic Bomb, which advocates this thesis, you'll find tons of footnotes (he is very good about that).

Here's the kind of thing that Alperovitz cites — a diary entry from May 14, 1945, of Henry Stimson, the Secretary of War, on a conference he had with General Marshall:

I told him that my own opinion was that the time now and the method now to deal with Russia was to keep our mouths shut and let our actions speak for words. The Russians will understand them better than anything else. It is a case where we have got to regain the lead and perhaps do it in a pretty rough and realistic way. They have rather taken it away from us because we have talked too much and have been to lavish with our beneficences to them. I told him this was a place where we really held all the cards. I called it a royal straight flush and we mustn't be a fool about the way we play it. They can't get along without our help and industries and we have coining into action a weapon which will be unique. Now the thing is not to get into unnecessary quarrels by talking too much and not to indicate any weakness by talking too much; let our actions speak for themselves.

OK — it's clear that Stimson, in this discussion, is thinking about Russia, and the atomic bomb plays a role in what he imagines the future action on diplomacy to be. Is that evidence of the idea that the bomb was dropped to intimidate the Soviets? Alperovitz things so, in line with a lot of other evidence. Other historians (including myself) think this perhaps overstates it a bit — Stimson was only one player in this decision (though an important one), and this was only one discussion he had, and it is not really directly about the question of whether to use the bomb. I think it is clear that if you read the volume of Stimson's thinking, he has a lot of things going on in his head. But if you buy the Alperovitz argument, this is one more piece of evidence. There is no "smoking gun" where people wrote down, "bwah hah hah, we did it to scare the Soviets!" But you would not expect there to be.

But to answer your question — the people who make this argument do have plenty of primary sources. The question is their interpretation. All primary sources require interpretation; there is no "raw data" worth speaking of. This is not unique to those people who make this argument; the people who disagree with them (myself included) have our own primary sources, our own interpretations, etc. To get a full sense of the evidence for it, read Alperovitz's book. But to be able to really make sense of whether the interpretation is plausible, you must either read many books, or read a meta-review (J. Samuel Walker has written pretty fair ones in my opinion) by another expert.