The guild leaders should be good citizens and help the emperors legally appointed governors to fulfil their duties in the best manner possible.
The governor, not being a local, might not know all the important people and issues. The guild leader should make sure that all the information he has about trustworthy and untrustworthy people, current and potential issues and ways to solve or prevent them is easily available to the governor. Naturally in a discreet and inoffensive manner, so that the governor won't feel like he is being imposed upon. But if a person affiliated with the guild happens to be talking with a person affiliated with the governor and the guild has helpful information about the topic it should be supplied in passing. And there is no harm in guiding such discussions towards pressing concerns.
To maximise the contact point the guild should also do its very best to help the governor connect with all the right people of his province socially. Find what interests the governor, his family and friends might have, arrange events catering to such interests and invite the interested people. It saves the governor and his inner circle the bother - and considerable expense - of trying to arrange such events themselves. This will allow them to fully focus in their duties to the empire and allow them easily and naturally access to all the information you think they should have. And more time they spend with reliable and trustworthy people, less time they will have to be exposed to untrustworthy people telling lies and falsehoods.
Naturally the help should extend beyond personal interests if needed. If the empire is really in decline and the central government is losing its grip, the governor might at times find himself in situations where the official resources available to him are simply insufficient to what must be done. At such time it is the civic duty of the guild and its leaders to help out behind the scenes with their own resources. Why should the soldiers keeping peace and order have to wait for their pay just because the imperial treasury is mired in corruption? The guild has money. Why should people starve when the guild can arrange imported food? Why should trade suffer from bad roads or piracy when the guild has resources to help the governor deal with it.
And more you help the governor, more time he will have for, not just enjoying his personal interests, but also to focus on the important things only he can do. Such as dealing with the imperial bureaucracy and making sure their inefficiency and corruption does not derail the smooth and efficient manner things work in the province thanks to the cooperation of all responsible parties.
Now in the unfortunate case the deterioration of the empire should continue, at some point there will no longer be any benefit for accepting the new governor the emperor sends. That is fine - all the soldiers and administrators already know the guild will pay their wages, just like they have always before. The people will know that the guild will take care of any issues that might arise, just like they always did.
For that matter it will probably not even be necessary to defy the empire by refusing the governor they send. If the ability of empire to maintain itself becomes compromised, it makes perfect sense to withdraw from some provinces in order to save resources and focus on securing the core of the empire. Obviously you do not want to leave a vacuum behind and it would be good to leave the province in hands of a reliable and cooperative party that can be trusted to keep that border safe. Typically the guild would be given control of the province for some period of time in exchange for a cash payment. If things get better the empire can reclaim the province. If not they will need the next payment more than another province to defend.
Even if control can't be bought outright. The empire should be willing to accept status quo of guild controlled provinces as long they are offered enough money, are treated with respect, and the cost of taking the provinces back by force would be impractical.
This establishes a safe, peaceful and mutually beneficial model for control of the provinces moving from the empire to the guilds. Nominally all the provinces would be part of the empire, of course. The Holy Roman Empire lasted a long time beyond the fall of Rome and even fall of Constantinople.