At the risk of advising you in matters you may already be familiar with, if you have little information on the Grandparents, then try backing up to the Parent's generation.
It may cost a little, but you can send for birth/death certificates for those of that intermediate gerneration that have passed away.From those you may be able to find an accurate first name and an idea when Laura and her husband may have died.Living relatives may be able to identify where they are burried.If you can find this and visit, dates can ususally be obtained from the grave markers (I've made several family trips "visiting dead relatives").
This kind of information will help if you try to send for a death certificate for Laura and/or her husband.Death certificates often include the names of parents and country/place of origin.Depending on your local laws, these documents are either kept by the locality/county or by the state government.
For the intermediate generation you can obtain their parents' full names from the Social Security applications.For contemporary relatives, you may sometimes find the person of interest in the Social Security Death Register (available online from several genealogy web sites for free) and obtain their SSN.With this information you can order images of the SS-5 registration forms from the Social Security Administration for deceased persons.
Hope I haven't lectured you too much.Again, good luck.