That would depend on one's definition of "normal." What's normal for one couple may be quite abnormal to another. In any event, it is very possible for people with Aspergers and High-Functioning Autism to develop an intimate relationship WITH THE RIGHT PERSON (i.e., someone who will learn about the disorder and make any necessary adjustments in relating to the Aspergers partner). In some cases, that “right person” may be another individual with the same condition who understands and has the ability to cope with the idiosyncrasies of another person with Aspergers.
Some of the barriers to relationships include a sort of “extended adolescence” or maturity issue in adults with Aspergers. This can mean that the individual marries later in life and lacks the ability to have solid relationships until they are older than the average person.
One of my Aspergers clients recently stated that he feels that the relationship with his wife is challenging, in part due to his overwhelming need to focus on his obsession of choice. He feels that he lacks a strong interpersonal connection and has to make a conscious choice to put his focus on his wife, to the exclusion of his desired focus of choice. He is accustomed to being solitary, and he finds it difficult to concentrate with others around him, including his wife.
Relationships do take a lot of work when one partner has Aspergers. The social skills required make relationships challenging for adults with Aspergers, particularly if diagnosed with it in adulthood. In fact, the divorce rate among "Aspies" is higher than in other groups of people.
Interventions (e.g., marital counseling) can work well if the therapist understands the unique features of Aspergers as it affects relationships.
We don't know statistically how many Aspergers adults develop "normal" relationships or how many find themselves unable to relate to a partner in an interpersonal and intimate way. But we do know that those with good communication and social skills have a better chance to succeed in a relationship than others.