This year (5780/2020), due to the coronavirus pandemic, there will likely be some Minyanim on Shiba Asar Be’Tammuz consisting of people who are not fasting. Patients who are sick with the virus, and patients recovering from the virus who still experience weakness or other symptoms, are certainly exempt from the fast. And given that people now are praying in small Minyanim, it is very possible that there will be some Minyanim consisting entirely, or mostly, of people who are not fasting.
Normally, on a fast day, if a Minyan does not have at least ten men who are fasting, then the Torah reading of "Ve’yehal" is not read. This ruling appears in the Mishna Berura (566:14). This year, however, Shiba Asar Be’Tammuz falls on Thursday, when in any event we read the Torah. Therefore, Hacham Bension Abba Shaul (Israel, 1924-1998) ruled that if even just six men in the Minyan are fasting, this suffices to allow reading "Va’yehal," and if fewer than six men are fasting, then they read the regular weekly Torah portion, which in this case would be Parashat Pinhas, as they would if Thursday was not a fast day. Of course, this would apply also when a fast day falls on Monday – the other weekday when the Torah is read.
This is applicable only at Shaharit, when the Torah would be read even if it weren’t a fast day. At Minha, however, if there are fewer than ten people fasting, then the Torah is not read at all, even on a Monday or Thursday.
If ten men are fasting during Shaharit, and thus "Va’yehal" is read, then somebody who is not fasting may not be called for an Aliya. In fact, the Shulhan Aruch rules that if on a fast day the only Kohen in the synagogue is not fasting, then he is asked to leave the synagogue, and three Yisraelim receive the Aliyot. According to the Magen Abraham (Rav Abraham Gombiner, 1633-1683), when a fast day falls on a Monday or Thursday, somebody who is not fasting may receive an Aliya to the Torah, since the Torah would be read even if that were not a fast day. Others, however, disagree, and so Hacham Bension rules that even when a fast day falls on a Monday or Thursday, when "Va’yehal" is being read, somebody who is not fasting should not receive an Aliya.
Summary: If a fast day falls on Sunday, Tuesday, Wednesday or Friday, then the section of "Va’yehal" is read only if there are at least ten men in attendance who are fasting. If a fast day falls on a Monday or Thursday, then at Shaharit, "Va’yehal" is read if at least six men are fasting, and if fewer than six men are fasting, then the weekly Torah portion is read. At Minha, even on a Monday or Thursday, ten people who are fasting must be in attendance for "Va’yehal" to be read. Whenever "Va’yehal" is read on a fast day, only those who are fasting may receive Aliyot to the Torah.