It's extracted from the seeds of grapes! so no it is not classed as nut
It's a commonly used base oil in massage and aromatherapy so you should find it easily available.
The 'plain massage oil' from Aston and Fincher may be fine, but you still need to know what it contains as plain massage oil means nothing.
You need to use a vegetable based oil and there are others - coconut, jojoba, evening primrose and safflower are all vegetable based, however, grapeseed, being a popular massage oil anyway, will be easier to source.