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Firstly:
It is not permissible to draw images of animate beings, whether the depiction is realistic or distorted, which is what is called cartoons, because of the general meaning of the evidence which indicates that it is haraam to draw and depict animate beings.
An-Nawawi (may Allah have mercy on him) said in Sharh Saheeh Muslim: Our companions and other scholars said that making an image of an animate being is emphatically prohibited and is a major sin, because this stern warning is issued against it in the hadiths. That applies whether it is included in something that will be handled without respect or otherwise; making it is haraam in all cases, because it is trying to match the creation of Allah, may He be exalted. It is all the same whether it is on a garment, a rug, a dirham (silver coin), a dinar (gold coin), a penny, a vessel, a wall or anything else. As for depicting trees, camel saddles and other things that do not show animate beings, that is not haraam. This is the ruling on image-making.
With regard to keeping anything in which there is a depiction of an animate being, if it is hung on a wall, or it is on a garment or turban that is worn, and the like, which are matters that are not regarded as disrespectful handling, then it is haraam. If it is on a rug that is stepped on, or a pillow or cushion and the like, that are handled with disrespect, then it is not haraam, but does it prevent the angels of mercy from entering that house? There is a scholarly discussion concerning that, which we will mention below, in sha Allah. In all these cases, there is no difference between that which has a shadow (three dimensional) and that which does not have a shadow (two dimensional). This is a summary of our view on the matter. Something similar was mentioned by the majority of scholars among the Sahaabah and Taabi‘een, and those who came after them. This is the view of ath-Thawri, Maalik, Abu Haneefah and others.
Some of the salaf (early generations) said: It is only that which has a shadow (three dimensional) that is prohibited, and there is nothing wrong with images that do not have a shadow (two dimensional). But this is an invalid view, because the curtain with images to which the Prophet (blessings and peace of Allah be upon him) objected was undoubtedly blameworthy, although its images did not have shadows. That is in addition to the other hadiths which spoke of images in general terms and are applicable to all images. End quote.
Secondly:
What appears to be the case is that it is not permissible to share these drawings in chat rooms, because that is spreading and propagating something haraam. Allah, may He be exalted, says (interpretation of the meaning):
“And cooperate in righteousness and piety, but do not cooperate in sin and aggression. And fear Allah; indeed, Allah is severe in penalty”
[al-Maa’idah 5:2].
Muslim (969) narrated that Abu’l-Hayyaaj al-Asadi said: ‘Ali ibn Abi Taalib said to me: Shall I not send you on the same mission as the Messenger of Allah (blessings and peace of Allah be upon him) sent me: Do not leave any image without erasing it or any high grave without levelling it?
The Prophet (blessings and peace of Allah be upon him) cut down the curtain that contained images, and tore it up.
If a person is commanded to erase images, how can he keep them and care about them and share them?
As for the notion that this is helping to convey the idea, it is obvious that means of da‘wah (calling people to Islam) must be in line with Islamic guidelines, and free of anything that is haraam.
And Allah knows best.