In fairness to Pemberton, Johnston's so-called Army of Relief didn't provide him with the support that it was supposed to.
From what I remember of the Vicksburg Campaign by the time Johnston had assembled an Army of the size he was happy would be able to make some significant impact on Grant's Army, rather than just be an annoyance to him, and was prepared to attack Grant and was moving to do so Pemberton surrendered.
It can be argued that Johnston's progress in Mississippi was hampered a lot by his arguements with Davis and his poor working relationship with Pemberton and as such he wasn't as efficent as in creating an army as he should have been but that is not entirely his own fault as his authority in bringing together the scattered Confederate forces in the Mississippi area was never really established as it should have been.
Because Johnston was not prepared to attack Grant unless his attack could have meant something in terms of impact on the Union Army and effect on the campaign he was not going to attack until he had enough men to make such kind of impact. He was planning an attack on Grant at Vicksburg but believed that such an attack on the Union Army could only succeed if Pemberton were to attack from Vicksburg at the same time and that the final goal of such an battle would be to get the Army of Mississippi out of Vicksburg and into an area where it would be usable again. Johnston also wanted to give tactical control of the proposed battle to Pemberton as he was not famillar with the area and was not as aware of the placement of the Union Army as Pemberton was.
Pemberton felt that Johnston should come and break the seige with his force while Pemberton protected the city and that Johnston was unreasonable in suggest that Pemberton take tactical command of the proposed battle. Pemberton also dissagreed with the final target of such a battle as he believed that the end goal should be the relief of Vicksburg not the removal of the AoM from it.
The arguement between these two over this proposed battle to relieve Vicksburg/save the AoM resulted in Johnston's Army of about 31,000 stalling in its advance towards Vicksburg at the Big Black River for too long and convincing Pemberton that he had to surrender. If Pemberton had been more willing to accept Johnston as his commander and followed his orders then this may not have happened but similarly if Johnston had been more open to Pemberton's side of the arguement and accepted its merits and attempted to change his plans accordingy it might have prevented the same situation happening.
The fact is however Johnston was not properly established as the top ranking Confederate commander in Mississippi, Pemberton thought he could change or simply ignore Johnston's orders, neither Johnston nor Pemberton agreed with the main objectives of the campaign and Davis, when he got involved, didn't help sort the problems at all but only helped undermine Johnston authority and further split apart the fragile working relationship between the two Confederate Commanders in the Theatre.
Compared to the united front that the Union forces showed under Grant its no wonder the Confederates lost Vicksburg, lost Mississippi and lost the Army of that State.
Bringing the