An Act of Henry VII’s (note Henry 7), in first Parliament in 1485 made unauthorized hunting in private forests a felony. It was punishable by death if the offence was committed at night or if the poachers had disguised or obscured their faces to prevent themselves from being identified; but if it was done in the daytime and without a disguise, it was only a trespass, punishable by fine or imprisonment.

There was one exception to the legislation. The authorities were alarmed at the damage caused by rooks and crows, who not only did great harm to crops but also damaged the thatch on the roofs of cottages and barns. An Act of 1533 stipulated that every parish must keep nets for catching rooks. Anyone was entitled to enter land without the landowner's permission in order to destroy rooks; they would not be liable for damages for trespass even if permission had been asked and had been refused.

So Shakespeare would have been able to get onto the Charlecote estate in 1583 but he would not have found any deer. Landowners were not allowed to keep deer other than in a deer park that had to be legally ratified. This did not happen at Charlecote until 1618.
There would have been plenty of other animals such as rabbits and hares and Thomas Lucy hated poachers. He would have had warreners patrolling his grounds. So Shakespeare might have been caught poaching, but not for deer.

As we will see in a moment, the legend was quoted by Nicolas Rowe in 1709 in the biography of Shakespeare he included in his complete edition of the Works of Shakespeare. But there are references to this legend even earlier than that such as by Archdeacon Richard Davies in 1688 who states that Shakespeare was often caught poaching and Thomas Lucy had him whipped and even imprisoned until eventually he fled to London.
It was inconvenient for the legend that there were no deer at Charlecote so the location was shifted to the adjoining estate at Fulbrook which was owned by Lucy and did have a deer park. The problem is one of dates. Shakespeare was ostensibly caught poaching in 1583 but the Lucys did not own Fulbrook until AFTER Shakespeare had died.