Hi Cdmshrs,
These are their meanings in the dictionary:
1. In successive and alternating order.
So in this case, to be better understood:
☛ when my turn came
2. Meanwhile, separately from the rest.
It's often used in written language but not in spoken language, and the most widely used form is "a su vez" since books are usually written in the third person.
Possessive varies with the person and number, so that we will say:
• Yo, a mi vez
• Tú, a tu vez
• Él/ella/ellos/ellas a su vez.
• Nosotros, a nuestra vez
• Vosotros, a vuestra vez
I sincerely hope this helps you, Cdmshrs! Have a nice day!