Yours in Progress,
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Bartolomiej, my experience in this area (mostly with nonprofit and volunteer organizations) reinforces Brittney's and Peter's thoughts! A few perhaps-minor additional thoughts follow:
-the multiple sites are both a blessing and a challenge: It's challenging to exercise leadership across many miles and time zones, and each site is likely to develop its own variant of the company culture (and sometimes even a literal counterculture). (I suspect but am not sure that this phenomenon may be weakened by extensive cross-site collaboration, but would be very surprised to see it go away entirely.) I see the multiple sites however as primarily a blessing in a small growing company, in that you can focus both climate assessments and leadership efforts on small, often-distinct teams, rather than being overwhelmed by a series of very large subordinate organizations. Hence, you have an excellent near-term opportunity to shape the growth of the company along with its culture.
-I think the advent of better synchronous collaboration technologies such as desktop VTC and peer-to-peer video chat gives you a great opportunity to "manage by walking around" at much lower cost than face-to-face, giving you many inexpensive opportunities to both sample the climate and offer impactful leadership. I suspect though you'll still have to do as much F2F as the budget allows, especially given your particular leadership portfolio.
-I'll cede the floor on climate-survey and other methodology tools and work to Brittney, Stefan, and others: Although I've crafted and successfully used these in the past on a limited basis, I count myself very far from expert in that area.