Can you use a reaction card on your opponent's drawing phase?

No, you cannot use a reaction card during your opponent's Draw Step.

According to RiftJudge FAQ #2886 and RiftJudge FAQ #2021, you do not receive priority during phase changes or steps like the Draw Step. You only receive priority to play cards or activate abilities when your opponent performs an action that creates a chain (such as playing a spell or unit) or during a showdown.

In summary:

• Phase changes and steps (like the Draw Step) do not provide a window for you to play reactions.
• You can only play reactions on your opponent's turn if there is an active chain or if you are in a showdown and have focus.