Should the verb "impact" be always followed by "on"?

Solution 1:

Not necessarily. The sentence you've quoted is perfectly correct as well.

Here are some sentences I've dug up where impact isn't followed by on:

Both events negatively impacted her life.

from Merriam-Webster

The book discusses the impact of Christian thinking on western society.

from Cambridge Dictionary

...the potential for women to impact the political process.

from Collins Dictionary

As far as I can see, your definition of impact (verb) can be used in four different ways: (from Collins Dictionary):

VERB + on/upon

as in

Such schemes mean little unless they impact on people.

VERB on/upon noun

as in

The reduction in the number of days that Parliament sat would impact on the quality of its work.

VERB noun

as in

...the potential for women to impact the political process. (mentioned before)

verb-link ADJECTIVE

as in

Trading is being increasingly impacted by the current recession.

Hope I've helped!

Solution 2:

Your first example is not idiomatic. If you use "impact" as a verb, the object of the verb (the thing impacted) needs no "on". If you use "impact" as a noun, however, you need the preposition.