Early spring can be okay if the tree isn’t waking yet, but scions are usually cut dormant in mid/late winter and stored. Then they’re usually grafted dormant to a rootstock that has the sap just starting to flow. That way you’re getting sap push from the rootstock through the scion as the graft tries to take and callous. You don’t want the scions to blow through their stored energy before the graft takes or they’ll start to dry and die out.
If you’re doing chip or T budding it’s a little different and you can do that any time during the warm season or even use semi-green wood with well developed buds and chip bud those in the late summer. That’s where you slice an individual bud off and graft it to the already growing plant stock you want.
I haven’t done any chip budding yet, but it’s done often with peach trees. I may attempt a few next year on my guardian peach rootstock I’m growing out with some Contender buds. Most of what I’ve done is with whip & tongue or cleft using whole dormant scion wood.
With fruit and nut trees, there’s differing times of the season for grafting certain fruit. Some fruit need warmer temps for the grafts to heal and callous or else they are more likely to fail. Apples and pears are easy peasy and are usually done early…but you can do them must of the warm season. Pawpaws and persimmons are later in spring as temps get more consistently near 80°. Many nut trees are later and can be difficult to get “takes”.