The verse occurs in this chapter of the Srimad Bhagavatam. Here is what the sons of Rishabha tell Nimi (the solar dynasty king whom I discuss here):
I consider that one whose intelligence is constantly disturbed by his falsely identifying himself with the temporary material world can achieve real freedom from fear only by worshiping the lotus feet of the infallible Supreme Lord. In such devotional service, all fear ceases entirely. Even ignorant living entities can very easily come to know the Supreme Lord if they adopt those means prescribed by the Supreme Lord Himself. The process recommended by the Lord is to be known as bhāgavata-dharma, or devotional service to the Supreme Personality of Godhead. O King, one who accepts this process of devotional service to the Supreme Personality of Godhead will never blunder on his path in this world. Even while running with eyes closed, he will never trip or fall. In accordance with the particular nature one has acquired in conditioned life, whatever one does with body, words, mind, senses, intelligence or purified consciousness one should offer to the Supreme, thinking, “This is for the pleasure of Lord Nārāyaṇa.”
Note that the verse has a slight difference from verse you gave:
kāyena vācā manasendriyair vā
buddhyātmanā vānusṛta-svabhāvāt |
karoti yad yat sakalaṁ parasmai
It says samarpayet tat - "he should offer that" rather than samarpayami - "I offer that". I'm not sure how it was changed into first-person form, or how it was incorporated into Hindu rituals like Sandhyavandanam. Perhaps it's originally from some other scripture and is merely being quoted by the Bhagavatam. It could be from some Pancharatra text, given that it's based on the notion of Karma Yoga.
By the way, the Sri Vaishnava sect (of which I'm a member) uses another verse called the Sattvika Tyaga for much the same purpose:
bhagavan eva svaniyamya sva-śeṣa-bhūtena maya
sva-ārādhana-eka-prayojanāya idam (name of the karma)
svasmai svaprītyai svayam eva karayati |
The auspicious deity, exerting control on himself, using me (his other part) as an instrument, himself effects , with his pleasure/ worship being the only purpose.
I'm not sure where this verse comes from either, but again the Pancharatra texts (which I discuss here) would be a safe bet.