Can I use a no data plan SIM card in a Galaxy S3

If the SIM card from the current phone fits, go ahead and use it.
AT&T does not allow smart phones on non data plans as they still use data. You have the option to turn data off at the account level if a data plan is forced on the line.

Note: if you are on the family share plan, she will be eligible for a line discount of $15 or $25 a month, depending on your family data plan. Multiply by 24 months and decide if it's worth entering another 2 year contract when a basic phone at full retail is less than $200. Then consider just replacing the phone at retail now. In 2 months, the discount will reduce her line fee.

http://www.att.com/cellphones/lg/Xpression.html#sku=sku5990223
Full retail is $170
Loss of discount is $360

I don't believe a "no data plan" sim exists, either the line has data or it doesn't. If the line is on a legacy non-mobile share plan (such as Family Talk), then using a smartphone on that line requires a data plan and a data plan will automatically be added once a SIM card is put into it.