Q:

Why are eunuchs allowed to ride the cart to Vaes Dothrak?

In A Game of Thrones the following paragraph indicates that, on the ride to Vaes Dothrak, a cart is used to transport eunuchs, cripples, women giving birth, the very young and the very old:

Khal Drogo had offered him a place in a cart the next day, and Viserys had accepted.
In his stubborn ignorance, he had not even known he was being mocked; the carts were for eunuchs, cripples, women giving birth, the very young and the very old. That won him yet another name: Khal Rhaggat, the Cart King.

Why are eunuchs allowed to ride the cart?

A:

They are not  considered "men" within the full context of the word. 
As you can see, the rest of the people mentioned are all weak: cripples, women giving birth, the very young and the very old. All of them are weak and most likely not able or allowed to ride a horse, therefore the Dothraki would consider eunuchs equal to cripples (which they are, in a way).
The Dothraki traditions are a little different from our own.
At Daenerys' wedding, she notes that the male Dothraki would mate with whoever, whenever and wherever they wanted. So a male Dothraki without the ability to mate, would be considered less of a man.
Just like @HorusKol said in the comments, this is a way to show how low their status is.

