To the OP:
Fertilizer is adding nutrients to the soil that the soil is lacking. You need a soil test to know what is deficient.
Vertical mulching often involves fertilizer, but doesn't have to. The point of vertical mulching is to improve soil texture.
Injections: Are you talking about trunk injections? The place for those is if you need to get something into the tree quickly and the soil is inhibiting that. The only time I use those is in high pH soil and you have significant iron or manganese deficiencies in the leaves. I'd HIGHLY recommend a tissue sample before you do those. If, for example, you assume iron deficiency, but it is in fact manganese deficient, you could push the tree over the edge as those to "compete" with each other. This is a "bandaid", and you need to start working on soil improvements or you are wasting the client's money.
Best time of year - spring or fall when there is good moisture in the soil. For fertilizer, if you put that in when it is dry, the salts will pull moisture out of the roots. If you try to vertical mulch in dry soil life is more difficult.