Frank, could some Genoese crossbowmen have participated in the battle?
Its entirely possible, as the Teutonic knights brought many western mercenaries and "Guests" to back them up in this war.
The battle of Grunwald was huge - historians argue about the specific numbers, but its mostly accepted that the Teutonic side fielded some 21-27 thousand men, while the Polish-Lithuanian side brought some 29-39 thousand men. So in total there were as many as 50 000 - 66 000 men on the field that day (for comparison, at Azincourt there were some 21 000 -30 000 men in total). A company of 50 or even 200 genoese crossbowmen could easily get "lost in the crowd" and thus is not explicitly emntioned by any period sources.
So they may have been there, but in small numbers in comparison with the total amount of men on the field. From what we know, the Teutonic order had plenty of its own foot crossbowmen.
During the battle of Grunwald, the only mercenaries we hear of (from contemporary sources) are:
1) Mercenary Knights from Westphalia "under a banner of two red arrows crossed on a white background".
2) Mercenary Knights from Switzerland, who had a white wolf on a red background as their banner.
3) Knights from "Lower Germany" (unspecified if theya re "guest-volunteers" or mercenaries)
4) Knights from Rheinland (unspecified wether they are mercenaries)
5) Knights from Meissen (unspecified, but the wording of the text seems to suggest that they were volunteers or official regiment of the prince of Meissen)
5) Unspecified mercenaries who had a banner with a checkkerboard of red and blue panels.
Both from the chronicle description, as well as the looks of their respective banners, it seems taht they were all heavy mounted noblemen. The Teutonic state was a very "urbanized" one in comparison with most powers of the period. It had a number of rich and populous cities. These provided a lot of spearmen and crossbowmen armed with "goverment issue" weapons and armor. It seems the Order had more than enough infantry and looked for armored mounted knights amongs mercenaries. So if there were any mercenary crossbowmen, they were likely a very small minority.