To paraphrase Pickett, I think Grant had something to do with Vicksburg. His campaign was brilliant, so it's hard to blame the disaster on Johnston or Davis.
That said, I do not think that Jefferson Davis was faultless. Joe Johnston ordered Pemberton to abandon the city and to join forces with him (Johnston). Johnston regarded the city as a trap. Davis, however, encouraged Pemberton to retain the city at all costs. If I recall correctly, Davis did not notify Johnston of his advice to Pemberton. Davis thus committed two errors. He gave terrible advice, and he did not tell his overall field commander that he was giving it.
Thus Pemberton was in effect caught between contradictory orders. Pemberton was not a gifted man, and his instinct was to retain the city; he did not recognize that Johnston's order was clearly correct. (In fairness, it may be unreasonable to have expected Pemberton to disregard the clear wishes of the president, his commander in chief.) Thus, Pemberton wound up losing both the city and the army.
As for troop levels, I'm not sure that I blame either Davis or Johnston (I assume the earlier post referring to Bragg meant to refer to Johnston). Davis had already transferred 20,000 troops from Bragg to Pemberton in early December 1862, which may well have spelled the difference between victory and perceived defeat at Stones River. After the scare in late 1862 (which resulted in the troop transfer mentioned above), Pemberton became confident that Grant could do nothing and delivered optimistic reports that would have given Davis no reason to consider additional transfers even if they had been feasible. After Stones River, Bragg was facing Rosecrans in the Tullahoma area and had no troops to spare. At all events, even if Pemberton had had more troops, it's doubtful that he would have used them in a way that would have made a difference.
By April 30, when Grant began crossing the river, it was probably too late to transfer troops from Bragg in time. If I recall correctly, there was no way to transfer troops quickly; it probably would have taken at least three weeks to get troops from the Tullahoma area to the Jackson/Vicksburg area, and again that assumes that Bragg had troops available. Grant's final campaign, however, basically took only a little over two weeks. By May 17, when the Battle of the Big Black River occurred, the game was essentially over. Once Pemberton was bottled up in the city, it became a matter of time.
There was one other option: transfer of troops from the trans-Mississippi. Perhaps someone else can comment on whether this would have been feasible and effective.