There are a number of reasons why pumpkins and squash will die or abort when small.  Incomplete pollination is one possibility. Pumpkins, squash, melons, and cucumbers all rely on insect pollinators to transfer pollen from the male flowers to the female flowers.  If this is not done well or completely the female flowers will begin to develop what looks to be a pumpkin but then abort that flower/developing fruit because it wasn't pollinated.  Doing what you can to promote pollinators to your garden will help.  

Unfavorable growing conditions are another reason why young developing fruit fall off or rot on the vine.  Inconsistent soil moisture and conditions that are too wet are two examples of growing conditions that cause fruit to abort or rot.  Provide consistent moisture for plants and avoid any situation where conditions stay too wet or too dry, especially while fruit are forming.  

Pumpkins and squash have a number of disease and insect issues that may also contribute to the fruit rotting.  The one that would most likely cause the issues you are seeing is blossom end rot. Blossom-end rot is a physiological disorder that occurs on tomatoes, peppers, eggplant, and squash. On pumpkins and squash, the blossom end of the fruit begins to rot and within a short time the entire fruit has rotted. Blossom-end rot is caused by a lack of calcium in the developing fruit. In most cases, there is no need to apply calcium to the soil. Try to maintain an even moisture supply by watering once a week during dry weather. Also, do not over-fertilize plants. Uneven moisture supplies and excessive nitrogen inhibit calcium uptake.

Best of luck!

Happy gardening!

--Aaron