The quick answer is that “exclusive” on its own probably does not convey enough information to be understood clearly as an indication of an exclusive provisorship, so if you're set on this construction you’re probably better off using one of the “exclusively” options.
Let’s start with a look at each phrasing you’ve thought of, and then I’ll suggest an alternative.
All spare parts shall be exclusive from [company name].
As I mentioned above, “be exclusive from” isn’t universally intelligible as establishing an exclusive vendor.
All spare parts shall be bought exclusively from [company name].
This one works well, but I agree with Jim that “purchased” sounds better here. All I can say is that it seems like a better match for the formal tone of the sentence. This option is the most exact and clear.
All spare parts shall be provided exclusively by [company name].
This one is also quite clear, and has the advantage of de-emphasizing the cost by not mentioning these compulsory transactions as such. This also detracts from its exactness, however, because “provide” could be interpreted as “given for free”.
All spare parts shall be bought/provided exclusively by/from [company name].
Simply too clunky. The main problem is probably that it involves two options without necessarily conveying which one of each pair goes with which one of the other.
If you don’t necessarily need to phrase it this way, you could use “exclusive” as an adjective describing your company (instead of using it to describe the parts). Then you’d write it something like this:
[Company name] shall be the exclusive provider of spare parts.
I've never heard or used "busy traffic" in this context. Rather, the roadway is what is busy with traffic. So I would use: "The traffic is usually heavy during rush hour," which would make the opposite "light traffic."
You should use intentions. "Are" is conjugated for the plural case, requiring the associated noun intention to be plural as well.
But beyond noun-verb agreement, intentions is still the best choice. If the singular form were used, it would mean that all road users share a single, unvarying intention. This is unlikely to fit the intended meaning of the sentence. The plural form allows for different intentions between different road users, as well as for a particular road user having different intentions at different points in time.