As Ignacio has said, the い-form (or the [連用形]{ren'yōkei} form) is used with に to indicate a purpose.
特別なビザをもらいに行った means you're going somewhere, and your purpose for going there is getting a special visa.
特別なビザをもらって行った, however, is quite incorrect. Indeed, ～て-form + いく can be used to make a compound verb that means "beginning from a certain point, to be in the process of gradually doing something". For instance "大きくなっていく" approximately means "to grow larger and larger". There's also "大きくなってくる" which would translate into the same thing in English, but has a different point-of-reference. With ～ていく, the process (of growing larger, for instance) starts at a certain point (pointed by the verb's tense) and goes on from there. With ～てくる, on the other hand, the process starting point is unmentioned, and it is viewed as progressing towards a certain point which is partially indicated by the verb's tense. For example, if X is the point of time we refer to:
人数が増えていった。 X ----> number of people grows ---->
人数が増えてきた。 ----> number of people grows ----> X
Note that X isn't necessarily the starting point or the ending point with either of these constructs. It just serves as the point of reference from which we look at it - in other words, ～ていく means we're looking forward and ～てくる means we're looking backward.
However, the sentence 特別なビザをもらって行った doesn't fit into this pattern, for two reasons. First, it's rather hard to imagine "getting a special visa" as a gradual process. Second, you only rarely find いく and くる in this sense (of auxiliary verbs that are used to indicate a gradual process) written in kanji. The standard is to write them in hiragana in such case. So 特別なビザをもらって行った does look plain wrong to my eyes.
