Well first of all its doubtful Hannibal or Carthage, even with support could've properly laid siege to Rome. Rome was a massive city, with extensive fortifications and a large garrison. Sieges of cities as large as Rome or Carthage were exceptionally rare in classical antiquity. It required huge amounts of manpower and engineering skill, time to adequately siege the city for what could be years, even then you still might not be able to take it.

Lets take the siege of Carthage as an example. Rome amassed a huge army, along with a second force of Numidian allies, and a full fleet. It amassed huge resources logistically for the campaign that probably could not be replicated by most other states around at the time. It continually reinforced this army with new manpower. By this time Carthage had been reduced to being a rump state. While still a powerful city itself it had lost the vast majority of its former territory and power and could count on no forces from other parts of its state to come and relieve the siege. This is compounded by the fact Carthage had no navy at the time and days before the siege was forced by the Romans to give up all of their weapons and ammunition. Despite this the siege lasted for 3 years, involved several roman defeats and tenacious fighting from the Carthaginians, coming very near to failure several times. Things were only restored by Scipio Aemilianus. Essentially taking a city as large as Carthage or Rome was near impossible unless you had virtually unlimited resources,time and manpower in which to conduct a siege.

Hannibal's plan was through either capture, or more favourably defection, remove the various allies and subjects of Rome until it was forced to accept favourable terms. Even Hannibal never planned on sieging Rome, we can see this in his treaty with Philip V were they both expect Rome to be around after the war. It should be noted this is pretty similar to what the Romans did to Carthage in the very same war. Remove its allies and conquer enough territory to where it is forced to accept peace. The Romans never planned on besieging the city either during the second Punic war.

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Now onto why didn't Hannibal get the support he needed to win the war? Well there are a couple of reasons.

1. The Carthaginians fundamentally had a different concept of war to the Romans. The Romans fought until one side was totally defeated, Hannibal himself was shocked when the Romans refused to even speak of terms after Cannae, by all rights they had lost. Under any other circumstances, Carthage would and should have won. They certainly would've gotten peace out of any of the Hellenic Kingdoms had they handed them a defeat as serious as even one of Cannae, Trebia or Trasimene, let alone all of them and the other defeats Rome suffered in the war. However the Romans realized that you are only beaten when you admit defeat, and that unique ability, at least of this Roman generation, to just keep taking hits won them the war.
2. The war was not just confined to Italy and the Carthaginians sought both to defend their own holdings and take advantage of the Roman loses to achieve victory on other fronts. Syracuse and Sardinia both requested Carthaginian help against Rome and so Punic invasion forces were sent there. Troops had to be sent to protect Spain, even the troops in Spain had to pop over to put down a Numidian revolt. This took a heck of a lot of Manpower, and while Carthage had great deals of it even she could not match Rome. Mercenaries and client levies take a lot of time to be recruited and mustered, and sending troops to one front means neglecting another.
3. As I mentioned above Carthage had a different philosophy of war to Rome, this included being less aggressive. Rome was an extremely aggressive nation in war, especially its commanders. To quote Adrian Goldsworthy "While its debatable whether Rome went to war more often than her contemporaries, she certainly did so with greater enthusiasm and effiecncy". Hoyos and Goldsworthy both mention that the Romans simply had a more aggressive take to war than the Carthaginians, and emphasize the continued aggression and decisiveness of the Roman commanders. Therefore during the war the Carthaginians generally reacted rather than being proactive in the war effort. For instance as mentioned men were only sent to Sicily and Sardinia at request, when really these should've been aggressive moves made earlier.
4. The poor performance of the Punic armies and Commanders. Simply put soldiers were sent to Sicily, Sardinia, Spain etc, because the Carthaginians were losing there, and not in Italy. The performance of every Punic Commander in the war who did not have the surname Barca was frankly abysmal in the extreme.The performance of Hasdrubal and Mago barca was reasonably good, despite their defeat in Spain by Scipio and later their defeats in Italy. Mago had almost won the battle at Insubria until his fatal injury, while Hasdrubal was undone by some brilliant Roman maneuvering both in campaign and during the battle that Hannibal would've been proud of. After that however things get pretty ugly. The expedition to Sardinia was pitiful, and the less said about it the better. The performance of Hasdrubal Gisgo was poor, and while he has some excuse in facing Scipio Africanus, his skill still left much to be desire, particularly with his foolishness in the African campaign. A quote from Fabius Maximus' son describes Hasdrubal as "a general who showed his speed chiefly in retreat". The performance of the other Punic commanders in Africa was equally poor, twice getting tricked into rash cavalry attacks, losing all of the cavalry that was their precious advantage against the Romans. Then we get to the Sicilian campaign. Were the performance of the Carthaginian commanders was probably the worst generalship of the entire 3 wars. Himilco was exceedingly coy, failing to take advantage of his greater numbers and support from the city to attack Marcellus, then he wasted his next campaign in camp, then he left the city and did nothing, then he messed up catching the reinforcements sent to Marcellus, then his army got plague. His replacement Hanno decided to fight a battle without his cavalry Commander the liby-phoenician Mettones (possibly some racism involved) due to jealousy, as Mettones had been achieving a number of cavalry successes against the Romans. Mettones cavalry promptly dipped out of the fight, the carthaginians still fought and got massacred. Hanno then replaced Mettones with his own son in an egregious bit of nepotism, at which point Mettones switched sides and let the Romans into Akragas (the armies winter quarters). Meanwhile Bomilcar spent most of his time evading Roman fleets rather than attacking him despite on atleast 3 occassions having superior numbers. Goldsworthy describes Bomilcars actions as "timid in the extreme". Hoyos is even harsher calling Himilco " one of the most 2nd rate commanders in Carthages history" and calls Bomilcar "Carthages worst admiral in history". This is already getting too long but the Punic perfomance in Italy as well outside Hannibal was wanting.
5. Alright rant over. The Carthaginians did try however to get reinforcements to Hannibal, and frankly the idea that Carthage was reluctant to send him reinforcements is simply untrue, and most contemporary scholars don't believe it. While Hanno the Great did disagree with reinforcing Hannibal his voice seems to have been the minority. By the second war, while the naval forces may have been equal, Carthage no longer held the advantage over Rome in naval supremacy, and the Romans may even have held the advantage, certainly the Carthaginian commanders were reluctant to risk their navy. Carthage lacked ports close to southern Italy from which to launch fleets. Rowing is very taxing, and storage space on warships is limited, most ancient fleets without transports couldn't sail for more than a week. Hannibal similarly lacked a large port from which he could be supplied, with the Romans managing to hold onto the citadel at Tarentum, and his failure to take Neapolis. This made reinforcement by sea difficult. However Bomilcar did manage to deliver 4,000 Numidians, some war elephants and cash to Hannibal at Locri. On the overland route both Hannibals brothers attempted to launch invasions of Italy to help Hannibal. Hasdrubal marched across the Alps and successfully got across before being defeated at Metauras as I mentioned previous. Mago landed in Genoa and fought a war in Northern Italy with the Romans for 3 years before his injury, and the recall of his army to defend Carthage.