I know it's silly to refer to Deadliest Warrior in some capacity here but some friends and I at Deadliest Fiction Wiki (basically fanon version of Deadliest Warrior that has greater merit over ScrewAttack's Death Battle) have looked at using the English Yeoman Archer and the thing that bothered me was that there was a remark that the Yeomans were active during the reign of Richard the Lionheart... who they thought was Richard III. Now I know there was probably a misnaming there because I'm sure as hell that "The Lionheart" was the nickname of Richard I not Richard III.
But I am a bit curious as to whether or not Yeoman archers existed before the 14th Century and the Hundred Years War. I've tried to look for myself but got nowhere. Maybe a historian here might do better?
This is the kind of question that either has a very long answer, or a pretty short one, and unfortunately I don't have time to write the really long one right now so hopefully the shorter one will do.
To start with, you are correct that Richard I, famous commander on the Third Crusade, is "The Lionheart" while Richard III, who died at Bosworth about 300 years later, is not.
There's a lot to unpack around the idea of the "Yeoman Archer" in late medieval English armies, which is where the extra length could come from, but if we simplify it to the idea of a semi-professional military archer then the answer becomes a bit simpler. The short answer is that no, the semi-professional archer didn't really exist during the reign of Richard I - it's largely a result of late medieval warfare and in particular changes in how armies were recruited which better enabled people to make a living by fighting in wars. That said, the bow certainly existed before then and so did archers, so if your question is how likely is it that Richard I had archers in his armies then the answer is very likely. Archers were very common for most of medieval warfare and especially in crusader warfare, so Richard definitely had them in his armies.
One extra detail that is probably relevant is that Richard I was far more famous for his love of the crossbow than for his opinions on the longbow. On two separate occasions on the Third Crusade, during the Siege of Acre in 1191 and while coming to the defense of Jaffa in 1192, Richard personally shot a crossbow in war and his armies generally seem to have had a fairly large number of crossbowmen (although records for army composition at that time are a lot vaguer and less consistent than for late medieval armies).
The tl;dr would be: Richard almost certainly did not have semi-professional archers in his armies, and while he did have archers he would have had primarily crossbowmen.
For further reading I recommend the following: