Nesting subordinate clauses immediately one after the other

Solution 1:

With the correct punctuation, both constructions are appropriate:

This is especially vital for technology companies, such as Google or Apple, that, while they are in fierce competition with each other, strive to create islands of cooperation and creativity within their organisation to create a climate conducive to innovation.

Makes essentially the same point as:

This is especially vital for technology companies, such as Google or Apple, that strive to create islands of cooperation and creativity within their organisation to create a climate conducive to innovation, while they are in fierce competition with each other.

With the commas in place, such as identifies Google and Apple as specific examples of technology companies that strive to create islands of cooperation and creativity.

The restrictive relative clause: that strive to create islands of cooperation and creativity within their organisation to create a climate conducive to innovation, is embedded in the independent clause, and it identifies what kind of technology companies are being discussed. It is also a matrix clause, because an adverbial subordinate clause is embedded in it.

The conjunction while embeds an adverbial subordinate clause within the restrictive clause. The subordinate adverbial clause modifies strive with the concurrent state of being: are in competition. A subordinate clause can be positioned before or after the main verb of the matrix clause, depending on the desired emphasis.

While communicates that strive to create is concurrent with are in fierce competition, so the order of the phrases has no impact on the logic of contrasting them.

The primary function of [while] as a conjunction is to indicate that two separate clauses occur at the same time.

Deciding which construction makes your point more effectively is a matter of style, which ELU doesn't address, but your climate metaphor has potential. Since an internal climate of cooperation is being contrasted with an external climate of competition:

This is especially vital for technology companies, such as Google or Apple, that cooperate in a climate of creative innovation within their organisations, while competing fiercely against each other.

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