Implementing Oauth1.0 from Apex involves passing a few parameters in the required format. Apex does support making a Oauth1.0 HTTP callout. Below is a screenshot from Postman, notice that the authorization has a bunch of token's and secret's.  Some are optional and we will see below how to pass them through Apex.

Consumer Key - Required

Consumer Secret - Required

Access Token - Optional

Token Secret - Optional

Signature Method - Required, usually it is HMAC-SHA1

Please note that the Access token and the Token Secret are optional, if you do have a token secret then the consumer secret will have that appended with a '&' symbol. For example,

Access Token Required => Consumer-Sercret&Token-Secret

Access Token not required => Consumer-Secret

Note that, every time the endpoint changes (you add new parameter values) you would have to sign it explicitly by calling the signRequest method.

References: https://www.salesforcelwc.in/2019/10/lookup-in-lwc.html

Below is a reusable custom lookup component. The standard lookup component usage is advised, but if you need to bypass the current user's sharing and security or you need more flexibility then use this custom approach. This code is adapted from the above reference.

DEMO: https://forcetree-developer-edition.na147.force.com/s/custom-lookup

STEP 1:

Create the Apex Class Lookup, and the LWC component customLookup

STEP 2:

Implement the customLookup component created above, as shown in the example below.

This post shows how to use SOSL to fetch records and display using LWC.

Demo: https://forcetree-developer-edition.na147.force.com/s/datatable-sosl