However, after the auction ended the buyer contacted me asking to pay cash on pick up. 

I told the buyer that I can not offer pick up but if they wanted to, they can come and pick it up from me.

The buyer then responded that they refuse to pay the shipping charge, and that local pick up was never specified. 

I never specified pick up was available at all.

I proceeded to tell the buyer that I will refund the shipping difference and received a response saying that they have contacted ebay.

Can ebay do anything in regards to suspending my account, etc?

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I'm a bit confused here.

Are you saying that although you never offered pick up in the ad, you agreed they could come and pick it up from you if they wanted to save on postage costs?

But that they now don't want  to do that because the ad never told them they would have to, but they don't want to pay the shipping charge either? What a hide they have!!

The ad would have told them the shipping charge before they bid.

What did they think was going to happen, that you might drop it off to them for nothing?

If they aren't happy with the postage cost and also refuse to pick up the item, I am not sure I would be asking for a cancellation, I would put in an unpaid item dispute. That way, if they don't pay and they lose the dispute, they can't go and give you neg feedback, or if they do, you can get it removed.

If it makes you feel any better, I am trying to sell a bedroom suite on gumtree at the moment, pick up only, and you would be surprised at the number of people who say they would like to buy it, but they live 50km away (with toll ways along the way, mind you), will I deliver? And not a word of paying a cent for delivery costs.

There are some cheeky people out there!!

I feel like saying, yes, sure, I'll pack it in the boot of my little corolla and bring it straight over.