Nowadays, we often see the word impact being used as a verb.
It has been a verb since the 17th century.
1601 P. Holland tr. Pliny History of the World II. xx. xxi. 73 The seed of this hearbe remooveth the tough humours bedded in the stomacke, how hard impacted soever they be.
1677 T. Gale Court of Gentiles: Pt. IV iv. Proem 4 Ideas or notions impacted on the mind.
Should the verb "impact" be always followed by "on"?
No.
1897 T. C. Allbutt et al. Syst. Med. III. 835 A stone-like mass..which had become impacted in the lower ilium.
1929 ‘Seamark’ Down River vi. 172 Something impacted with a soft thud against Lingard's temple.
1962 F. I. Ordway et al. Basic Astronautics v. 201 The Soviet Lunnaya Raketa was launched early in the afternoon of September 12,
1959 and impacted onto the Moon's surface just after midnight on September 14, Moscow time.
The meteor will impact the Earth soon.
The cuts will inevitably impact on service delivery
is incorrect.The cuts will inevitably **have an** impact on service delivery
is the correct grammar. Also,The cuts will inevitably **have a negative** impact on service delivery.