We will begin with the smaller problem – namely, using a bicycle on Shabbat, irrespective of the place in which it is being ridden. When bicycles became popular, many poskim discussed their use on Shabbat, and almost all forbade it for one or more of the following reasons: 1) Uvdin d’chol – It is a weekday-like activity. Among other examples, it is a mode of transportation that takes people to many places for many purposes, including non-Shabbat-appropriate ones.[1] 2) Bicycles are prone to require repairs that the rider often takes care of, which could cause him to transgress Shabbat prohibitions.[2] 3) One might ride beyond the techum Shabbat (boundaries of permitted travel outside the city).[3] Although Rav Yosef Chayim of Baghdad[4] dismissed these issues (some say he later changed his mind), the consensus of both Ashkenazi and Sephardi poskim and the broad minhag is to forbid use of bicycles on Shabbat.
One might claim that when a vehicle is ridden by a non-Jew for a Jewish passenger, one can be much more lenient, and a rickshaw thus operated might be too uncommon to have a clear minhag. While we do not totally dismiss this approach, the logic is somewhat difficult. After all, the Jewish participant is taking part in the weekday-like activity, he could (help) fix the rickshaw if necessary, and he could be taken out of the techum Shabbat.
These problems do not seem to apply to a man-driven rickshaw, which is conceptually similar to a baby carriage or a wheelchair. While this type of “transportation” is also subject to discussion,[5] the accepted consensus clearly permits it.
However, there is another problem of using a rickshaw in some places, including all of India: Pushing or pulling a carriage through the streets, by foot or by bicycle, is a form of carrying.[6] Since there are no eiruvin in India, this is an additional problem on Shabbat. There are some mitigating factors, however. Carrying or pulling a person who is able to walk by himself is permitted on the level of Torah law because of a principle called “chai nosei et atzmo”;[7] it is forbidden only Rabbinically. Although in our case the driver/puller is also transporting the rickshaw that the Jew is sitting in, we assert that the carriage is just an extension of the person.[8] (To rely on this leniency, however, the Jew must not be carrying anything that he could not take into the streets himself.[9]) Another factor is that there is no Torah-level violation when the carriage is large enough to constitute its own “private domain.”[10]
Even after these factors are taken into account, there is still a Rabbinic prohibition to carry a person outside. You asked, however, about a case in which a non-Jew is doing the work. The Rav Pe’alim[11] permitted one to use the services of a non-Jew pedaling a rickshaw that was of the right dimensions, but only if the Jew needs this in order to enable him to do a “big mitzva.” In general, one is allowed to ask a non-Jew to perform a Rabbinic prohibition in order to facilitate a mitzva.[12]
Many disagree with the Rav Pe’alim’s application of these principles (including the issues of bicycles) when the Jew himself is involved, even if the non-Jew pedals.[13] However, it is possible to consider using this leniency for a person-drawn rickshaw in the case of a specific mitzva that one cannot get to in another way, which is unusual for a healthy person.
If one pays a non-Jew for a job, this may sometimes make it permitted for him to do work on Shabbat, because it is considered as though he is doing it for himself to earn money. However, this does not apply if the Jew needs the non-Jew to do the work (that is forbidden for a Jew) specifically on Shabbat,[14] and all the more so when it will be done with the active participation of a Jew. In this case, various forms of hinting will also not be effective. We understand that giving the non-Jew money in advance may be technically necessary for him to be willing to provide the service.
The rabbinic community might want to discuss having non-Jews transport Jews in rickshaws on Shabbat for extreme cases, like a special mitzva or for crucial needs of the sick or infirmed. However, under normal circumstances, this system is forbidden, especially in an area that does not have an eiruv.