We recently had a married couple as seder guests and found out that they are in the process of a reconstructionist (non halachic ) conversion. They seemed interested in coming to a future shabbat meal. May they be invited? I don't want to give the impression that an orthodox person would normalize non halachic conversion and they seem unaware of the difference in conversion status by denominations. It also seems inappropriate to suggest an orthodox conversion, as that is prosyletizing. Should non halachic conversion candidates be regularly welcomed to future Jewish events in an orthodox home ?
I've been at Shabbat meals in Orthodox homes where Jews with non-Orthodox conversions, and for that matter gentiles who were studying for conversion, were present. The hosts knew the status of the guests and took the necessary precautions -- using mevushal wine, not asking one of them to lead any of the b'rachot on behalf of the group, not counting them for the zimmun, and maybe other things I haven't thought of. One host told me some years back that he does this as a matter of kiruv; sometimes his guests had never actually experienced the joy of a traditional Shabbat evening, and if it inspires them to look more closely, that's a benefit.
There are some different halachic concerns about inviting non-Jews for meals on Yom Tov (h/t msh210). This answer is specifically about Shabbat.
You need to explain the fact that by Orthodox Jewish standards, and traditional Jewish Law (halacha), they are not considered Jews and will not be considered Jews when they complete their Reconstructionist conversion. However, there should be no problem with inviting them to the Shabbat table.
They cannot, however, lead any of the brachot, and the man cannot count for the zimmun. Also, using mevushal wine would be advised.
They should be aware of the fact that they are not counted as Jews so they will not feel snubbed by you, since you will not offer them to lead any of the brachot and traditions done at the table. And, if they feel snubbed by you not giving them these honors because they are not counted as Jews halachically, then there is nothing you can do about that.
I suggest you read the other answers here as they provide additional information.
You can invite everybody, but you can explain friendly your point of view. It is not proselytism to show your orthopracticing family. If they are interested by a Shabbat meal with you, may be that they are searching their way.
After conversion, in the reconstrionist way, are people Jewish? This is a second part of the question, I have no knowledge about this specific group.