Should I take a look to see if there are the people who raise dogs ? Q: In the above sentence, can I replace "should" with "would"? thanks in advance!
"Should" and "would" are both grammatical but convey different meanings.
"Should he go to New York" asks whether it is desirable, advisable, or proper for him to go to New York.
"Would he go to New York" asks whether he is willing to go to New York.
Proper grammar depends on intended meaning.
What is not idiomatic is "the" before people. When "people" is used in the sense of "some persons among a totality," the definite article is not used. You should say "if there are people who raise dogs."
Let's first suppose the sentence is simpler.
Should I take a look to see if there is orange juice in the refrigerator?
That question is grammatical and makes sense. Then, replacing "should" with "would":
Would I take a look to see if there is orange juice in the refrigerator?
This does not make sense in the same way. From a basic point of view, it's incorrect. From a more advanced point of view, this is a rhetorical question. The person is requesting themselves to do something, or asking if they themselves will do something. Only they can answer.
Would you take a look to see if there is orange juice in the refrigerator?
Is a polite request to someone else.
Returning to your original sentence, "if there are people who raise dogs?" means "in general, are there people in the whole world who raise dogs?". However, since you said "look", you probably mean something more specific, such as "Should I take a look to see if the people who raise dogs are there?" Notice the re-ordering of words. Still, for that sentence, the same advice applies. "Would I" is basically incorrect, and changes it into a rhetorical question. "Would you" is alright.