Like the question says, do the scriptures say that we should clean our bodies with water every day? What happens if one forgets or does not feel like showering? is it classified as a sin?
Yes it is mandatory. In fact it is one of the 3 Pratahkrittas ( the 3 deeds that must be done by all at morning; pratah is morning ).
When the dawn arrives, one should, after duly performing the purificatory works (i.e., passing urine and excreta) and cleansing the teeth, bathe in the morning. (6)
Bathing in the morning is the purifier of the highly dirty body, having nine apertures, [and] passing [urine and excreta] day and night. (7)
Daksha Smriti, Chapter 2, Verses 6, 7
The same Smriti says one who does not bath at least once in the morning, suffers from perpetual impurity. So, he becomes unfit for performing any kinds of rituals.
Perpetual impurity is spoken of for all of them, who,- without bathing, offering oblations to the Fire and making gifts, partake of [their] meals'.
Daksha Smriti, Chapter 6
Some more verses from the same Smriti are given below:
Bathing has been spoken of as being threefold, viz. Nitya (daily), Naimitika (obligatory, as in the solar or lunar eclipse), and Kamya (having an end, such as the attainment of the celestial region in view;. (37)
A man, neglecting his own duties, and failing to bathe, offer oblations to the Fire, recite the Mantrams and make gifts, does not become [a householder] by [merely having] a son and a wife. (47)
So, daily bathing is one of the compulsory duties for everyone. Specially if its done at the time when sun rises or just before that time then its considered as even more auspicious.
Bathing in the morning, at the period of conjunction when the Sun rises, is equal to the [penitential rite of] Prajapatya in the destruction even of mighty iniquities. (11)
The book Dharma Bindu quotes Maharshi Vyasa as follows:
Vyasa opines: Praatah kaaletu sampraapte kritvaa chaavashyakam budhah, Snaayaannadeeshu shuddhaasu shoucham kritvaa yadhaa vidhi ||
A virtuous person would perform his morning routine and then take bath in rivers or running water
Further it quotes Atri Muni as follows:
Atri explains further: Asnaataasheemalam bhuktehyajapah puyashonitam, Ahuutaashee krimim bhuktehyadaataa vishamashnute/ Snaana mulaah kriyaa sarvaah shrutismrityuditaa nrinaam, Asnaatastu pumaannaarhah japahomaadi karmasu/ Praatar madhyaahnoh snaanam vaanaprastha grihasthayoh, Yatestrishavanam snaanam sakruccha brahma chaarinaam/ Sarvepi sakrit kuryuh Ashaktaochodakam vinaa, Saamardhyechaambu sadbhaave yathaa shastram hitat bhavet, Snaanam cha sarva varnaanaam kaaryam shochaparassaram ||
A person who is not bathed but eats food, that food is as bad as excretion, just as a person does not meditate and eats food that food is as horrible as consuming blood; food without performing Agni Homa is stated to eating worms and food without daana is to be considered as poisonous! Both Grihastaas and Vaanaprasthaas ought to take bath twice in the morning and evening, Sanyasis thrice and Brahmacharis on ‘sakrit’ basis or as when water is unavailable or not possible physically beyond the minimum a day. In fact, this clause of Sakrit is available to all, but subject to one’s own ability and shortage of water the escape clause is not applicable. In any case, the first bath in the morning is compulsory!
But, if one is ill then he can either do a rub/sponge with wet towel or take a warm water bath. Both are prescribed in the scriptures.