Thanks Sergey.
I've checked 'Allow inprocess', saved, closed SSMS, opened, relogin (windows auth.) and get error (see below). Also see my code below.

Script for allowing inprocess
USE [master]
GO
EXEC master.dbo.sp_MSset_oledb_prop N'VFPOLEDB', N'AllowInProcess', 1
GO


Error: (same as original):
Msg 7399, Level 16, State 1, Line 1
The OLE DB provider "VFPOLEDB.1" for linked server "Test" reported an error. Access denied.
Msg 7350, Level 16, State 2, Line 1
Cannot get the column information from OLE DB provider "VFPOLEDB.1" for linked server "Test".

I did not get the error when I used SQL Server authentication.

My code:
declare @z int
EXEC @z = sp_droplinkedsrvlogin 'mark', 'verinow\Mark'
print @z

exec sp_dropserver 'test'

declare @x int
EXEC @x = sp_addlinkedserver
@server = 'Test',
@srvproduct = '',
@provider = 'VFPOLEDB.1',
@datasrc = '\\mainserver\poy\Fox_2002\boNotify\Data'
print @x

select * from sys.servers where name = 'Test'

SELECT *
FROM OPENQUERY(Test, 'SELECT * from company')