In the U.S., this is largely a question of state law and while similar from state to state, it is not identical.
In Colorado, which basically follows the majority rule, there are a couple of justifications that could be available: (1) the use of force justified for defense of property, and (2) the use of force justified for a citizen's arrest.
In the case of defense of property, the rule is as follows:
Use of physical force in defense of property
A person is justified in using reasonable and appropriate physical
force upon another person when and to the extent that he reasonably
believes it necessary to prevent what he reasonably believes to be an
attempt by the other person to commit theft, criminal mischief, or
criminal tampering involving property, but he may use deadly physical
force under these circumstances only in defense of himself or another
as described in section 18-1-704.
Colo. Rev. Statutes § 18-1-706.
In the case of a citizen's arrest, the rule is:
A private person acting on his own account is justified in using
reasonable and appropriate physical force upon another person when and
to the extent that he reasonably believes it necessary to effect an
arrest, or to prevent the escape from custody of an arrested person
who has committed an offense in his presence; but he is justified in
using deadly physical force for the purpose only when he reasonably
believes it necessary to defend himself or a third person from what he
reasonably believes to be the use or imminent use of deadly physical
Colo. Rev. Stat. § 18-1-707(7).
Can the victim shoot the pickpocket and claim heat of passion as a
Deadly force is not authorized against a pickpocket, and shooting someone almost always counts (often by definition) as a use of deadly force.
The "heat of passion" defense only downgrades a murder charge to manslaughter and probably wouldn't apply in any case in these circumstances.
Can the victim grab, tackle, or punch the pickpocket? Can the victim
say, "Give back my wallet or I will punch you"? Does the answer vary
depending on which of the two people would be at an advantage in a
physical confrontation, based on size, age or gender?
The law doesn't answer these question at this level of specificity.
Whether the force used was "reasonable and appropriate" and was "reasonably believed to be necessary" are determined after the fact on a case by case basis by the finder of fact (i.e. the judge in a bench trial, or the jury in a jury trial).