How should a theravada centre approach the lay people for support? I was planning to visit a centre, so first I checked their webpage, I thought they are a lit too pushy when it comes to donations, asking for money in a very explicit way to pay their bills. OK, let me be fair, they have bills to pay and utilities can be expensive sometimes.

I have more experience with Mahayana (eventhough I follow Theravada now) and in the two centres I have been to, I have never seen similar thing, so it makes me wonder, why? Maybe their funding structures are different, more robust, I dont know...

Is there anything in the Vinaya that can help? What are the limits when it comes to asking for money?
My favorite essay on dana is Thanissaro Bhikkhu's "No Strings Attached:
The Buddha's Culture of Generosity".

In it Thanissaro Bhikkhu points out the main criteria Buddha himself set for right dana (AN 6.37):


  The donor, before giving, is glad; while giving, his/her mind is inspired; and after giving, is gratified. These are the three factors of the donor…
  
  The recipients are free of passion or are practicing for the subduing of passion; free of aversion or practicing for the subduing of aversion; and free of delusion or practicing for the subduing of delusion. These are the three factors of the recipients.


He then goes on to show that explicitly asking for donations goes against the above:


  In pressuring retreatants to give to teachers, it doesn't lead to gladness before giving, and instead sounds like a plea for a tip at the end of a meal. The frequent efforts to pull on the retreatants' heartstrings as a path to their purse strings betray a lack of trust in their thoughtfulness and leave a bad taste.


He mainly argues against putting hard dana amounts on dharma talks (making them paid lectures), but I feel it does apply to your case as well.

It's not clear what he proposes instead though, he does not really explain it at great details:


  The ideal solution would be to provide a framework whereby serious Dhamma practitioners could be supported whether or not they taught. ... a genuine culture of dana ...


I personally believe in membership fees. Those lay people who are associated with a given Dharma Center for a long time, could organize a sort of PTA club (with optional membership!), collect membership fees among themselves, and then donate them to the center.