How can I design this database?
Indexes don't have to be unique. Primary keys do. 

The purpose of a Primary Key is to uniquely identify a single row of data. If you don't have something that is naturally unique then add a column such as an identity column and define it as the primary key. It's standard practice, and will help you later if you need to update a row. 

If the table is being queried for analysis or reporting a single table is fine. Do some analysis on the types of queries you are performing and add indexes on relevant key columns. Good indexes will significantly improve query performance.

For example if you are looking for results for a single store, by adding an index on the store id, you could exclude the other 299 stores from the select. This reduces IO and speeds up the query. If you have years worth of data but are only looking for things that happened in the last week then adding an index to a date column may be a big help.

Look at your queries and see if there fields you are regularly filtering on. Start with those.

Ensure the db has updated statistics for the table and see if performance improves. 

The results you get will vary depending upon what you are trying to do. If you are trying to aggregate records (i.e total sales for each store) then the query may still have to read the whole table. 

Below I've added a script with a basic test example. 

USE [YOURDBNAME]
GO

SET ANSI_NULLS ON
GO

SET QUOTED_IDENTIFIER ON
GO

SET ANSI_PADDING ON
GO

-- Create Test Table. 

CREATE TABLE [dbo].[rewards](
    [id] [int] IDENTITY(1,1) NOT NULL,
    [fname] [varchar](50) NULL,
    [lname] [varchar](50) NULL,
    [tstamp] [datetime] NULL,
    [card_id] [int] NULL

) ON [PRIMARY] 

-- Add Primary key. This can be done at table creation. Or added after as shown below.

ALTER TABLE rewards ADD CONSTRAINT pk_rewards PRIMARY KEY NONCLUSTERED  ([id])

-- Add an index on columns that your query will use.

CREATE CLUSTERED INDEX [IDX01] ON [dbo].[rewards]
(   [card_id] ASC,
    [fname] ASC,
    [lname] ASC 
) ON [PRIMARY]

-- Create some test data. 

INSERT INTO [dbo].[rewards] VALUES ('Allan2', 'Aadvark2', getdate(), 2345) ;
INSERT INTO [dbo].[rewards] VALUES ('Billy2', 'Babo0n', getdate(), 2356) ;
INSERT INTO [dbo].[rewards] VALUES ('Chester2', 'Cheetah', getdate(), 2367) ;
INSERT INTO [dbo].[rewards] VALUES ('Doodoo', 'Dog', getdate(), 2376) ;
INSERT INTO [dbo].[rewards] VALUES ('Esme', 'Elephant', getdate(), 1235) ;
INSERT INTO [dbo].[rewards] VALUES ('Freddy', 'Fox', getdate(), 1239) ;

GO

-- run the following query repeatedly to generate volume. 
INSERT INTO [dbo].[rewards]
           ([fname]
           ,[lname]
           ,[tstamp]
           ,[card_id])     
     select [fname]
           ,[lname]
           ,[tstamp]
           ,[card_id]
           from [dbo].[rewards] ;

GO 

SET STATISTICS TIME ON
GO 

-- Test query. This shows that the card 1234 is use by 2 different customers. 

Select r.card_id, count(*) 
from (
    select card_id, fname, lname
    from [dbo].[rewards] 
    group by card_id, fname, lname ) r

group by r.card_id
having count(*) &gt; 1 ;

GO 

SET STATISTICS TIME OFF

GO 

-- I just tested this with 3,538,944 rows. And I get SQL Server Execution Times:  CPU time = 1904 ms,  elapsed time = 490 ms.
-- On repeated execution I can reliably get this under .5 of a second. 


Your results may vary depending upon machine setup and load, But I think you should resonably expect to get your query down to ~1 sec or less.