'Prosper', unlike 'strive', is a close synonym of 'thrive'. (Illustrations using 'prosper' are perhaps more accessible.)
prosper verb (used without object)
to be successful or fortunate, especially in financial respects;
thrive; flourish. [Dictionary.com]
Sentences such as
- The home of the famous Yorkshire Mixture, Joseph Dobson’s is a
fascinating story of a family owned business, which has prospered to
become one of the largest family owned confectionery manufacturers in
England. [Dobsons UK – our business]
where 'to become ...' is an adjunct containing additional closely connected (showing results / expanding on the matrix sentence) material, are quite acceptable.
'which has prospered to become ...' could be replaced by
'which has prospered greatly, becoming ...' or
'which has prospered greatly – so much so, that it has become ...'. ('greatly' added merely to lubricate)
is an example of the same king using 'thrive'.
So the sentence 'They prospered/thrived to become one of the largest ...' is by no means unacceptable (though I'd use the -ing form 'becoming' instead here).
However, the injunction 'Thrive' or 'Prosper' sounds very strange here, belonging in a religious or otherwise supernatural domain. One can't say that 'Thrive! Become an efficient and adaptive IT organisation!' is ungrammatical, but it's certainly verging on the outlandish.
It's quite likely, as I think you suspect, that 'Strive [= 'try your utmost'] to become an efficient and adaptive IT organisation', where strive (like try, attempt, seek) takes a to-infinitive clause thus forming a catenation, is what was intended, and that this is a malapropism.