I’m not entirely sure all Catholic altars do have relics nowadays, especially the ones that are constructed of wood and made to look similar to normal tables. However, Vatican II never said anything about doing away with the tradition.
Here are two sources of information about altars and relics: GIRM 298-302 and CIC 1235-1237. I’ve snipped out information that is irrelevant to the matter at hand:
GIRM 298. It is appropriate to have a fixed altar in every church … An altar is called **“fixed” if it is attached to the floor **so as not to be removeable; otherwise it is called "moveable."
GIRM 300. An altar whether fixed or movable is dedicated according to the rite prescribed in the Roman Pontifical; but it is permissible for a movable altar simply to be blessed.
GIRM 302. The practice of placing relics of Saints, even those not Martyrs, under the altar to be dedicated is fittingly retained.
Can. 1235 §1 The altar or table on which the eucharistic Sacrifice is celebrated is termed fixed if it is so constructed that it is attached to the floor and therefore cannot be moved; it is termed movable, if it can be removed.
§2 It is proper that in every church there should be a fixed altar. …
Can. 1237 §1 Fixed altars are to be dedicated, movable ones either dedicated or blessed, according to the rites prescribed in the liturgical books.
§2 The ancient tradition of placing relics of Martyrs or of other Saints within a fixed altar is to be retained, in accordance with the rites prescribed in the liturgical books.
So, long story short: it is “proper” and “appropriate” for every church to have a fixed altar, and fixed altars “are to be dedicated” (while moveable altars can simply be blessed), and the tradition of placing relics in the altar to be dedicated “is to be retained”.
My parish doesn’t have a fixed altar (as far as I know), though, so “proper” might be one of those loosey-goosey words. I would hope, though, that every fixed altar really does contain relics, since the tradition of doing so “is to be retained”.