Let us first look at the occurrences of words such as [gun]powder and pistol in the play as a whole. Falstaff says (Act V, scene 3),
I am as hot as molten lead, and as heavy too. God keep lead out of me, I need no more weight than my own bowels.
I have led my ragamuffins where they are peppered.
The scene is set during the Battle of Shrewsbury and Falstaff's ragamuffins are the soldiers he is in charge of.
A few lines below, Prince Hal enters the scene and says to Falstaff,
I prithee lend me thy sword.
Falstaff responds (emphasis added),
Nay, before God, Hal, if Percy be alive thou
gets not my sword, but take my pistol if thou wilt.
Unfortunately, what the Prince draws from Falstaff's "case" turns out to be a bottle of sack, which is more useful before than during a battle.
In Act IV, scene 2, Falstaff had described the men he had rounded up for the army "food for powder",
showing little concern for the lives of his men, but also providing evidence for the use of firearms.
And in Act V, scene 4, Falstaff also use the word gunpowder (in a metaphoric sense):
'Zounds, I am afraid of this
gunpowder Percy, though he be dead
The word "pistol" is also used elsewhere in the play, namely by Prince Hal in Act II, scene 4:
He that rides at high speed and with his pistol
kills a sparrow flying.
Based on this textual evidence, it is perfectly plausible that "lead" in "keep the lead out of me" refers to bullets.
It is much less plausible that it refers to something like the leaden dagger that Prince Hal mentions in Act II, scene 4 because lead is too malleable to be used for a dagger:
Thy state is taken for a joined-stool, thy golden
sceptre for a leaden dagger, and thy precious rich
crown for a pitiful bald crown!
So what about the historical accuracy? The use of gunpowder and pistols during the Battle of Shrewsbury (July 1403) sounds anachronistic. The Wikipeda article doesn't mention any firearms. This is not because the technology was unknown in England though. The Wikipedia article on gunpowder artillery mentions that the "pot-de-fer" was used by both the English and the French during the Hundred Years' War (1337–1453) and that
During the 1340s, cannon were still relatively rare, and were only used in small numbers by a few states.
(So this cigarette card in the collections of the New York Public Library is perhaps not entirely anachronistic. But the depicted shape does not match the image of the reconstructed pot-de-fer in the above Wikipedia article.)
The French started using the culverin in the 15th century; this weapon was "later adapted for naval use by the English in the late 16th century", i.e. long after the Battle of Shrewsbury.
The arquebus, however, did not appear in Europe before 1521, a century after the Battle of Shrewsbury.
What Falstaff's pistol might have looked like is less clear. Wikipedia explains that "the English word was introduced in ca. 1570 from the Middle French pistolet (ca. 1550)". So Shakespeare has Falstaff use a word that was not introduced into the language until roughly 170 years after the Battle of Shrewsbury. The term "anachronism" is fully justified here.