Is it permissible to seek assistance through experience – even if there is no Sharia-based evidence for it – in performing certain acts, such as reciting specific Surahs in special situations like losing something or suffering from an illness, and are there any Sharia-based controls for it if it is affirmed?

What you have mentioned falls under the science of the properties of the Qur'an, and scholars have differed on this matter: Some considered theخصصing specific verses or surahs with virtues or for fulfilling needs without proof to be an innovation (bid'ah), while others relied on experience to prove these properties. One researcher has clarified that the properties of the Qur'an have a basis in the Qur'an and Sunnah, and that the entire Qur'an is a healing and mercy for the believers.

There is no harm in reciting the Qur'an as a means of supplicating to Allah Almighty for the fulfillment of needs, as this is a permissible form of intercession (tawassul) through righteous deeds. The Prophet (peace be upon him) encouraged asking Allah through the Qur'an.