In general when it comes to the topic of,
Using language contained in the job posting and company website
It makes sense to carefully read and understand the language an employer uses on their website, job descriptions, career page, etc. It can provide you a glimpse into the culture of the employer and the specific type of person they're looking for.
Mirroring the same voice, vocabulary, level of complexity, and overall style in your correspondence (without going overboard or stepping outside the norms of formal communication) can help you show that you understand and fit the profile they're looking for.
To answer your specific question,
since it's on the career page, would it be okay to use it?
I'd say, it depends.
As you're acknowledging, "awesome" is a pretty informal slang word and wouldn't normally appear on a resume or career website. So - rather than focusing on that specific word, look at the bigger picture. Does this company portray a less formal culture through the rest of the language on the career page and in their job descriptions? If so, then yes, it's a great idea to use that word, and it'll be a specific nod to the fact that you're paying attention. If "awesome" seems a little out of place in a sea of otherwise-formal language, I would advise against it, but it's perfectly fine to mirror the sentence otherwise, as you're planning when you said:
I don't see a position, but I do have a
n awesome skillset to be a great addition to your team.
Just make sure you can back it up - be prepared to show in your letter and CV what your skills are, how you've solved problems similar to what they may be facing, etc etc. This sentence won't mean much on it's own of course, but can be a good intro and a way to show that you actually read their webpage.
And as a final thought, you need to filter all of this through your own personality. Don't try to be someone you're not - you want them to hire "you" not a character you make up to try to fit exactly what they're looking for. Overall, choosing a single word (or not) is a minor part of a bigger picture of matching yourself to the employer.