The petitioners believed that their impure status was a not a good reason to prevent them from participating in the Paschal rite, and that there was a legal basis for such a claim. The sources disagree regarding the specific reasoning given:
The people were requesting that they be able to participate in the Pesach rite since by nightfall, when the Pesach was to be eaten, they would already be pure.
Who were the petitioners? The questioners were people who had become impure on the eighth of Nissan. The seventh day of their impurity fell on the fourteenth, so that they were still impure at the hour that the sacrifice was brought, but by nightfall, when the Pesach was eaten, they would have been purified.
"וְלֹא יָכְלוּ לַעֲשֹׂת הַפֶּסַח בַּיּוֹם הַהוּא" – R. Yitzchak learns from the phrase "וְלֹא יָכְלוּ לַעֲשֹׂת הַפֶּסַח בַּיּוֹם הַהוּא" that it was specifically on that day (the fourteenth) that they could not bring the Pesach. The following day, however, they would have been able to.
What part of the rite did they want to participate in? The people did not expect to be able to actually offer the sacrifice (as they were clearly impure at that point), but were hoping that they could be counted with someone else's offering, and that they would eat of it themselves at nightfall.
Focal point of Pesach: sacrifice or eating? It is possible that the petitioners assumed that the focal point of the Peach was the act of eating,
4 and therefore thought that if they were pure for that aspect of the rite, it should not be problematic if they had not yet been purified when the Pesach was sacrificed. Hashem's negative response might be an indication that the sacrifice itself is a crucial part of the ceremony.
5 Uniqueness of Pesach – The Pesach is unique in that it is sacrificed in the afternoon of one day (14th), but only eaten at night, which is the next day (15th). This is what allowed the petitioners to suggest that there might be some halakhic leniency to allow them to participate.
Halakhic precedents – Or HaChayyim suggests that the people compared themselves to those who have touched an insect (שרץ) whose impurity only lasts for a day. In such a case, the impure are allowed to immerse on the fourteenth and then be included with someone else's sacrifice and eat at nightfall. The petitioners believed that the law might be the same for them.
When were laws of impurity given? This position might assume that only some of the laws of impurity had been given prior to our story. Thus, the petitioners might have been familiar with the laws of insects, but not all of those which related to touching a dead body.
6 As the full discussion of the red heifer and purification from the dead first appears in Bemidbar 20, it is possible that many details of the rite were first taught then.
Why wait until the 14th to ask Moshe? According to this approach, it is not clear why the people would wait until the last day to ask Moshe the law, given that they knew their status in advance. The Ritva suggests that the parallel to other laws made them overly confident that there would not be a problem.
7 Why is Moshe unaware of the law? As the people found themselves in a unique situation (there is no other sacrifice where it would be possible to be pure for part of the rite and not for a different part) it is not surprising that Moshe might not have been aware of the law.
Hashem's response: is the request granted? Hashem's response is somewhat difficult for this position. If the people's legal reasoning was wrong, why does Hashem not simply say so and refuse the request? The offer of an alternative date seems to have nothing to do with the specific arguments of the people and thus should have been announced regardless of this specific case (and not, as the text implies, as a reaction to it).
The people requested that they be allowed to partake in the sacrifice via an agent, since there was no need for them to individually perform the ritual and their impurity did not necessitate them to disassociate from the rest of the nation.
Who were the petitioners? According to Abarbanel the people making the request had come in contact with a dead body at some point in the week before the Pesach was to be sacrificed and would only be fully purified at some point after the fourteenth. It is even possible that they had first become impure on the very day of the offering (which would explain why they first asked about the law on that day.)
What part of the rite did they want to participate in? They were asking to be counted with someone else's sacrifice, but had no expectations of actually eating of it, recognizing that their impurity precluded doing so.
Legal reasoning – In contrast to other forms of impurity such as a metzora or zav whose impurity entails being sent out of the camp , one who is impure due to the dead is only excluded from the vicinity of the Mishkan itself. As such, the people thought that there should be no impediment in their joining the rest of Israel as they sacrificed.
Uniqueness of Pesach – Every Pesach sacrifice is brought by a group of people of whom just one acted as the representative to sacrifice. As such, there was room to believe that the impure could be part of the group as long as they personally did not actively sacrifice nor eat. This might explain the petitioner's emphasis on wanting to sacrifice "בְּתוֹךְ בְּנֵי יִשְׂרָאֵל", as part of a group of pure Israelites.
Focal point of Pesach: sacrifice or eating? The petitioners might have assumed that the sacrificial aspect of the rite was the most important, and therefore concluded that the fact they would not eat was not particularly problematic. Hashem's response might suggest that eating is just as important as sacrificing and therefore anyone who is excluded from the meal, cannot be included in the sacrifice either.
Why wait until the 14th to ask Moshe? According to this approach, it is possible that the people first became impure on that day.
Why was Moshe unaware of the law?
The petitioners claimed that a sacrifice which has a set time supersedes and pushes off the laws of impurity. Therefore they should be allowed to participate in the rite.
Who were the petitioners? Netziv does not specify and would likely say that they could have been anyone who had become impure via contact with the dead.
Legal reasoning – Since a sacrifice which has a set time is allowed to be brought even on Shabbat or in a state of impurity, the petitioners assumed that the Pesach fell into such a category and that their impurity would be deemed irrelevant.
"בְּמֹעֲדוֹ" – The people emphasize this since it was specifically because the sacrifice had a set time that they felt that they should be allowed to offer it even while impure.
"בְּתוֹךְ בְּנֵי יִשְׂרָאֵל"