As CTLAWYER’s family did, my paternal grandparents bought burial plots in the 1940’s and we added to those in the 1980’s. My maternal grandparents and most of their siblings bought plots in Old Montefiore Cemetery in the 1930’s through Adas Yisroel of the Lower East Side (aka United Hebrew Communities). My grandparents even bought their tachrichim while they were in their thirties.
There is no official time, but if one first waits to buy one when someone needs to be buried, the family will be paying a “premium price” for a grave(s).