Q. A woman ovulates during her 7 clean days and therefore cannot get pregnant. Is it permissible to use a method whereby the husband would produce his seed into a syringe and the woman would then insert it herself during her 7 clean days (while she is still a Nida as she has not yet gone to the mikva)?
A. The question can really be split into two separate halachic issues, the first is whether putting in the seed during the 7 clean days is permitted and if the subsequent child being born from this would be considered pogum (blemished) in any way. Secondly is how to obtain the semen from the husband.
It is important to note that there are ways in which you can postpone ovulation until after the mikvah. One way is to take Clomid (clomiphene citrate). Another way is to take estrogen. Obviously these treatments must have a prescription and not every doctor is familiar with using these medications for postponing ovulation so one should consult with a Rav or someone competent who can properly advise on these issues. Even so these medications don't always help.
Regarding the first problem of inserting semen into a woman during the seven clean days. The Taz in Y.D. 107 permits a woman to lay on her husbands bed (not whilst he is on it) during her nidah state and we are not worried about any semen on the husbands sheet making her pregnant. The reason being that the whole blemish in a child born from a nidah is in the actual illicit act of the cohabitating while she is a nidah. But if there is no prohibited act the child born is completely fine. R' Moshe Feinstein in Igrois Moishe E.H. 2: 18 goes with this Taz.
Regarding the second issue of obtaining semen, although there should not be a problem of wasting male seed because here the semen is not being wasted rather being used to make the woman pregnant, nevertheless R' Moshe Feinstein in the same place says that there would still be a problem of "niuf b'yad" meaning he may not use his hand to obtain the semen. R' Feinstein only permits him to obtain semen through mental fantasizing. Unfortunately, practically today that method doesn't work too often. Another way would be for the couple to have relations while the woman is not a nidah and then freeze the sperm until she ovulates. Freezing sperm has advanced in recent years whereas this was not possible at all in R' Feinstein's days now technology allows for super fast dry freezing that does not crystallize and ruin the sperm. Even so the fertility rate of frozen sperm goes down quite drastically (depending on which freezing method is used). Another option would be to do an EEJ or Electroejaculation, where they basically insert a probe and with electrical pulses caused an ejaculation while the patient is under general anesthesia. This method also has problems in the motility of the sperm produced. It is important to note that the Achiezer 3: 24 allows for the use of one's own hands in order the produce semen for this purpose. An interesting question would be if it would be permitted to use a medical vibrator according to R' Feinstein. Horav Yitzchok Berkowitz Shlit"a maintains that freezing sperm is the best method and R' Feinstein would not permit the medical vibrator.
Rabbi Dovid Bartfeld as revised by Horav Yitzchok Berkowitz Shlit"a
Horav Shlomo Miller's Shlit'a opinion is that a point should be made in that some Poiskim consider the born child as "pogum", although completely permitted. (Shiltey Haguiborim, on Ran Shovuos p.4). He does not think that Igrois Moishe would permit using a vibrator and also suggests the best method to be freezing the sperm.
Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit"a