Somatic embryogenesis (SE) has many applications in grapevine biotechnology including micropropagation, eradicating viral infections from infected cultivars, mass production of hypocotyl explants for micrografting, as a continuous source for haploid and doubled haploid plants, and for germplasm conservation. It is so far the only pathway for the genetic modification of grapevines through transformation. The single-cell origin of somatic embryos makes them an ideal explant for mutation breeding as the resulting mutants will be chimera-free. In the present research, two combinations of plant growth regulators and different explants from flower buds at two stages of maturity were tested in regard to the efficiency of callusing and embryo formation from the callus produced in three white grape cultivars. Also, the treatment of somatic embryos with the chemical mutagen ethyl methanesulfonate (EMS) was optimised. Medium 2339 supplemented with β-naphthoxyacetic acid (5 μM) and 6-benzylaminopurine (BAP—9.0 μM) produced significantly more calluses than medium 2337 supplemented with 2,4-dichlorophenoxyacetic acid (4.5 µM) and BAP (8.9 µM) in all explants. The calluses produced on medium 2337 were harder and more granular and produced more SEs. Although the stage of the maturity of floral bud did not have a significant effect on the callusing of the explants, calluses produced from immature floral bud explants in the premeiotic stage produced significantly more SEs than those from more mature floral buds. Overall, immature ovaries and cut floral buds exposing the cut ends of filaments, style, etc., tested for the first time in grapevine SE, produced the highest percentage of embryogenic calluses. It is much more efficient to cut the floral bud and culture than previously reported explants such as anthers, ovaries, stigmas and styles during the short flowering period when the immature flower buds are available. When the somatic embryos of the three cultivars were incubated for one hour with 0.1% EMS, their germination was reduced by 50%; an ideal treatment considered to obtain a high frequency of mutations for screening. Our research findings will facilitate more efficient SE induction in grapevines and inducing mutations for improving individual traits without altering the genetic background of the cultivar.