In some cases it is true; speculators can affect prices. They raise prices by buying goods and storing them, thus reducing the total amount available to be sold. They lower them by selling goods from their inventory.
As best I can tell by a little casual googling, world storage capacity for petroleum comes to less than two months output. At any given time, much of that is already in use. If, say, 80% of storage capacity is already being used, which I gather is a not particularly high figure, then the most speculators can do is to take off the market something less than two weeks worth of output—once. It is hard to see how that could have much effect on price for more than a short time.
There is, however, a cheaper way of storing oil. The reason to buy and hold oil is the expectation that you will be able to sell it in the future at a higher price. That is also a reason why a producer with limited supplies might choose to pump next year instead of this year. Storage capacity in that form is essentially unlimited; producers could choose to pump no oil at all and leave all of it in the ground. That suggests that, insofar as speculation is responsible for current high prices, it is speculation not by speculators buying oil and putting it in tanks but by oil producers leaving it in the ground today so as to pump and sell it tomorrow, or next year, or next decade. I do not know enough about the evidence on actual and potential output to guess how likely that is.
One of the odd features of the current political fuss over all of this is the widespread assumption that high prices due to speculation are a bad thing. That assumption seems to be shared by, among others, most of the people who argue that we are running out of oil and so should use less of it. If we are running out of oil that is a good reason to shift consumption from the present to the future, when it will be scarcer. Which is exactly what speculation, by producers or speculators, does.