I've worked somewhere like this and little by little I did turn it around.
The directors were massively over-worked and stressed and had no time at all for me. Weeks could go by where no emails I had sent with queries or requests would be answered.
My best tips are:
A weekly schedule sent to managers of work that I was doing with updates on outcomes or activities I couldn't complete due to lack of response etc.
Diary timetables for everything including meeting requests, which I scheduled with agendas issued.
Sending fait accompli type emails, e.g "I have seen that X department is unable manage email due to technical issues. I have contacted tech team and they recommend x,yz. Unless I hear otherwise by xxx date, I will instruct tech to resolve on xxx date at a cost of xxx." That way they only needed to comment if there was an issue.
It's soul destroying when either no one could give a shit or they are too busy to respond. You have my sympathies.
Ultimately, you either push and probe until you can shape things the way you want them, or you look elsewhere and then leave.
I'm glad I true and it did help, but I won't take I a dept/company like that again as the stress was not pleasant.