The short answer is yes, they can come in a curly form. There are several types of fresh noodle used in Rāmen, which can be classified mainly according to thickness and shape.
Noodles are classified in shape into the straight sutorēto-men (ストレート麺）, the curly chijire-men（縮れ面）, and the more rare flat hirauchi-men（平打ち麺） . With the exception of the flat type, these are further divided into several degrees of thickness: the extra thick gokubuto-men（極太麺）, the thick futo-men（太麺）, the middle thick chūbuto-men（中太麺）, the middle thin chūboso-men（中細麺）, the thin hoso-men（細麺）, and the extra thin gokuboso-men（極細麺）. The thin types are also further differentiated in the amount of water they have in their dough.
Which shape and thickness of noodle is used depends on the style of ramen and on the chef's preference. For example, hakata rāmen（博多ラーメン）, of the sorts served by Ippūdō in their Japanese stores, usually uses straight gokuboso type, but the same franchise uses chijire gokuboso type for their spicy rāmen and chijire gokubuto for their tsukemen. Likewise, yokohama tonkotsu-shōyu （横浜豚骨醤油）, also known as ie-kei（家系）, uses chijire-gokubuto.
The choice of noodle is related to many factors, but cooking time is an important one. In hakata rāmen, where the practice of topping up your noodles (kaedama／替え玉) is common, one has to be able to cook the noodle quickly, and hence the very thin noodle. Thicker types of noodle need longer cooking time (aprox. 10 min for gokubuto-men), and are only used by rāmen traditions that do not offer a noodle refill.
Another factor affecting the noodle type is the type of broth. Thicker broths ask for straight type of noodles, and thinner broths for curly noodles.
These are not rules etched in stone, however, as rāmen itself is a sort of creole food born from a mix of Japanese and Chinese cuisines. Some would say that there is not right or wrong in rāmen, and that anything is possible. However, what I have described above is what is customary and what you will find if you go to rāmen shops in Japan.
By the way, not all rāmen dishes in Japan use fresh noodles. There is a type of rāmen that originates from Chiba prefecture and is called takeoka rāmen（竹岡ラーメン）, which was originally cooked by part-timer old ladies hired from the neighbourhood, and uses dried (what you would call instant) noodles because they are easier to cook.