Can I make handmade invitation cards on a tight budget?
If you have plain white paper that is thick enough to make invitations cards, you can do the rest with free materials.

Cut and fold the paper into cards, then neatly write the invitation. 
Cut decorative forms like hearts, lollipops, bonbons or Hello Kitty heads and glue them onto the cards. You can cut these forms out of colorfull sheets of ads or catalogs or anything colorfull you have lying around.

Catalogs for clothes or furniture often have single items printed very big (like a carpet, windows drapes or a t-shirt). The only disadvantage is that the pages are often so thin you can see the backside through.

An alternative is wrapping paper. If it's the birthday of your little sister, you probably have a gift to wrap for her. Use some scraps of that paper to decorate the cards. Pieces of gift wrap ribbons add a special flair.

Depending on what you have lying around at home, you can also repurpose things like confetti, glitter, small beads or golden makeup powder do decorate the cards.

Take care not to add too much to the card. 1 - 3 decorations are enough to make the card look special.