Several types of cannons were used during the wars of the roses, obviously the longbow was very prominent during this time, did either side have access to and use hand-held guns at any point during the Wars of the roses?
Most likely. Contrary to some claims, field artillery drastically changed late medieval battle tactics, and were essential to understanding just went down at battles like Orleans, Castillon, and Tewkesbury.
Guns were developed on the Continent, apparently primarily within the French, Burgundian, and Ottoman arsenals. We know that gunners numbered in the five hundreds in the field armies that Edward VI and Henry Tudor brought from the Continent, and it is hard to believe these mobile armies had enough large cannon to need such a large group. It is more likely that many gunners were used to bolster the archers, firing smaller, armor-piercing weapons en masse to harass enemy men-at-arms.
We can presume that French mercenary gunners used methods analogous to French army practices. In the closing battles of the Hundred Years War, hand-held guns were relatively common in the royal artillery park, although tactically the effect of the culverins field-pieces may have been more important, because the culverins' range was farther than that of bowmen, giving the French a tactical edge over traditionalist English longbowmen tactics, and the Yorkist army an edge at the battle of Tewkesbury.
According to David Nicolle, a two-man handgun team, in which one man holds the nascent arquebus stock, and the other apparently aimed, was the common method. This is akin to how bazookas and panzerschreks were used in WWII, and reflects one basic fact. Early guns were very heavy, and the goal in battle is consistency. While it is feasible many gunners fought singularly, it seems that fighting in teams, either around a culverin or smaller gun, was the more general tactical method that best used the features of a heavy, slow firing weapon.