The term for when fruit is first starting to appear on a plant

Solution 1:

Set is the verb.

(Of fruit) develop from blossom:

'once fruits have set, feed weekly with a high potash liquid tomato fertilizer'

[OD]

The stage of fruit set follows flowering almost immediately, when the fertilized flower begins to develop a seed and grape berry to protect the seed.

^{Following fertilization, the fruit begins to form.}

_{Annual growth cycle of grapevines / Wikipedia}

Solution 2:

How about fruit, the verb?

As in,

The grapes are beginning to fruit

Solution 3:

I would say that the vines are budding:

bud, v.: To put forth or produce buds
bud, n.: A small protuberance on a stem or branch, sometimes enclosed in protective scales and containing an undeveloped leaf, flower, or leafy shoot.
_{source:Thefreedictionary.com}

As you can see, this does not necessarily refer specifically to fruit, but is widely understood to be when any plant starts to produce growths.

Alternatively, you could refer to them as being in bloom, in blossom, or blossoming, all of which refer to the flowers of the plant that produce the edible fruit.