How to use yup’s object.shape with typescript?

following the yup guide here you should change your code as follows:

import { object, SchemaOf, string } from 'yup';

interface BaseBootstrapSchema {  
  APPLICATION_NAME: string;
  LOG_PATH: string;
}

const bootstrapValidationSchema: SchemaOf<BaseBootstrapSchema> = object({
  LOG_PATH: string().required("The 'LOG_PATH' ENV variable is required"),
  APPLICATION_NAME: string().required("The 'APPLICATION_NAME' ENV variable is required"),
});

module.exports = (schema: SchemaOf<BaseBootstrapSchema>) =>
  new Promise((resolve, reject) =>
    schema
      .validate(process.env, { abortEarly: false })
      .then(() => resolve(true))
      .catch((error: any) => {
        if (error.errors.length > 1) {
          reject(new Error(`${error.message},\n${error.errors.join(',\n')}`));
        } else {
          reject(new Error(error.message));
        }
      })
  );

Actually you don’t need the const bootstrapValidationSchema at all, since you will pass it to the node module you are exporting, but I could’ve misunderstood what you are trying to achieve here.
Try it.