As @juanpethes mentioned, the best way to do this for the Google Search API is to add site:domain.com to your q param. For example: q=Coffee site:amazon.com OR walmart.com
Sample Playground link: https://serpapi.com/playground?q=Coffee+site%3Aamazon.com+OR+site%3Awalmart.com&location=Austin%2C+Texas%2C+United+States&gl=us&hl=en&newPara=as_sitesearch+chips
Since you mentioned shopping results, I should note Google Shopping does not support these site:domain.com type filters. Instead, you need to refer to the filters array that SerpApi returns in a request to the Google Shopping API. There are several options within filters including Stores which provide a few a stores you can filter results too.
These are limited to what Google Shopping offers for any given search unfortunately. You cannot filter for any arbitrary store you want. This is a limitation of Google that SerpApi cannot work around.
For example:
https://serpapi.com/playground?engine=google_shopping&q=Coffee
"filters: [
...
{
"type": "Stores",
"options":
[
{
"text": "Target",
"serpapi_link": "https://serpapi.com/search.json?device=desktop&engine=google_shopping&gl=us&google_domain=google.com&hl=en&q=target+coffee&shoprs=CAEYAioGY29mZmVlMgoIAhIGVGFyZ2V0WLfLH2AC"
}
,
{
"text": "Walmart",
"serpapi_link": "https://serpapi.com/search.json?device=desktop&engine=google_shopping&gl=us&google_domain=google.com&hl=en&q=walmart+coffee&shoprs=CAEYAioGY29mZmVlMgsIAhIHV2FsbWFydFi3yx9gAg"
}
,
{
"text": "Home Depot",
"serpapi_link": "https://serpapi.com/search.json?device=desktop&engine=google_shopping&gl=us&google_domain=google.com&hl=en&q=home+depot+coffee&shoprs=CAEYAioGY29mZmVlMg4IAhIKSG9tZSBEZXBvdFi3yx9gAg"
}
,
{
"text": "Sears",
"serpapi_link": "https://serpapi.com/search.json?device=desktop&engine=google_shopping&gl=us&google_domain=google.com&hl=en&q=sears+coffee&shoprs=CAEYAioGY29mZmVlMgkIAhIFU2VhcnNYt8sfYAI"
}
]
}
...
]
Use the serpapi_link for the store you want to filter by to get results only for that store. You just need to append your api_key to the request.
Disclaimer: I work for SerpApi.