Is Red Cedar suitable for making arrow shafts.

In my opinion, no.  Too soft and too brittle.

I agree, too soft and really hard to get clear wood. 

Eastern or Western red cedar? Two different trees.

Not familiar with eastern.  I was thinking of western when I responded. 

WRC is very straight grained but light in weight. I think it is strong but maybe too brittle. With a good straight grained board you could split a few out and try them.
ERC is actually a juniper, Juniperus virginiana.

We have Eastern Red Cedar, so that is what I need to know about. Thanks.

I think if you can find a good clean, straight grained piece of ERC it would make good arrows although they would probably be physically light.