Speaking from an American perspective; there are several branches of the Amish Order. The Old Order Amish (the ones that adhere to horse and buggy, no modern conveniences, etc.) are unlikely to have 'outside' converts, for a number of reasons, none of which, in my estimation, are racist.
Their way of life is exceptionally strict and bounded. Meaning, the work is hard and requires a great deal of native skill handed from one generation to another, as well as being content with ones location for the rest of their lives.
Secondly, most are of German descent and still speak a dialect of German language that seperates them from the rest of society. Both place restrictions on adherents as well as potential converts.
The more liberal branches of the Amish, easpecially the ones we see here in the U.S., such as Mennonites and Hutterites also have a well developed work ethic, but utilize moderninity readily. They also tend to inter-marry and do still speak German, but not as strictly as do the Old Order.
I've developed friendships with some of our local Hutterite colonies and do not observe much in the way of converts, but I have no doubt that they would be welcomed if the 'newbies' were willing to subject themselves to such a Spartan existence...