First consideration is the size. All the 'A Painting A Day' paints are on A4 as this is small enough to store in ring binders that most shelves can accommodate. It also means that every painting has a regular size. Doing larger pieces is fun and presents other interesting challenges due to it's scale. Whatever you use it is always good to think it through and make sure that if you are going to do a lot of them where you are going to store them.
Any paper can be used if resources are scarce. The thicker the paper the less likely it is going to buckle. It is possible to stretch the paper however as the object is to make things as easy as possible and not put unnecessary barriers in the way, a weight of at least 250 g/sq m cartridge paper is good and will not need any additional preparation before starting. If you choose to use a specialised paper then the weight can be less as there is a coating which prevents the buckling being so pronounced. The reason for opting for a multi-purpose paper such as Clairfontaine Multi-Technique paper is that is is relatively inexpensive and can be used for many mediums.
The cartridge paper is slightly absorbent which means that the paint dries a little quicker than those with a harder finish. There a papers which are dimpled and some which imitate the look of a canvas. For the purposes of A Painting A Day a smooth finish is the best option as it does not interfere with the texture of the paint itself. However the choice is yours. Play around with different papers and see which one suits you.
If you have lemons make lemonade
Perhaps there is not a supply of paper or you are on a budget, use whatever you have to hand. It is the process of painting which is the objective. If you only have printer paper no problem. It has been used before and will stand heavy texturing if it is handled gently. Cardboard is a great alternative and can be made from used packaging. If that is your only option then it will need an undercoat or primer to seal it. Any emulsion paint will do the trick.