"Fox kit" may exist, but I can't find "kit" listed in either the OED or the American Heritage dictionary in the sense of a young fox.
AFAIK the normal designation for a young fox is "cub", "Young fox" is, according to the OED, in fact the root meaning of that word.
1. orig. A young fox.
1530 PALSGR. 211/1 Cubbe, a yong foxe. 1552 HULOET, Cubbe or yonge ffoxe, vulpecula. 1575 TURBERV. Venerie 181 When you have taken the old foxes or badgerdes, and that there is nothing left in the earth but the yong cubbes. 1648 Hunting of Fox 13 His skin..when he is a young Cubbe is usually of a darker colour. 1880 Times 2 Nov. 4/6 No cub is he, but a full-brushed, high conditioned, dog-fox.
2. a. By extension: The young of the bear and of other wild beasts; also of the whale.
For the young of the bear, lion, etc. the earlier word was whelp, as in all versions of the Bible from Wyclif to 1611.
1596 SHAKES. Merch. V. II. i. 29 Plucke the yong sucking Cubs from the she Beare. 1683 BURNET tr. More's Utopia (1684) 13 The old Crow loves his Young, and the Ape his Cubs. a1687 WALLER (J.), Two mighty whales..One as a mountain vast, and with her came A cub. 1774 GOLDSM. Nat. Hist. (1776) II. 334 The lion, or tyger, have seldom above two cubs at a litter. 1823 SCORESBY Jrnl. 148 The smallest animals [whales] of the species, mere cubs or ‘suckers’. 1829 SCOTT Anne of G. ii, With the fury of a bear which had been robbed of her cubs.
(etc.) (c) OED.