See FrankerZ's comment, apparently this dispatch function is a standard Redux thing (I don't use Redux) and it doesn't return a promise, so it doesn't make sense to use Promise.all on it.
But answering the promises aspect of "how would I wait for getThis first": Just move it to the beginning of the chain:
return function (dispatch, getState) {
    dispatch(getThis())             // First this, then...
    .then(result => Promise.all([   // ...these
        dispatch(getThat(result)),  // (I've passed the result to all three, but...)
        dispatch(getThese(result)),
        dispatch(getThose(result))
    ]))
    .then(() => {
        let state = getState();
        dispatch({
           //dispatch
        });
    }).catch(function (err) {
        console.log(err);
    });
}
or without the dispatch, since apparently that's wrong:
return function (dispatch, getState) {
    getThis()               // First this, then...
    .then(result => Promise.all([   // ...these
        getThat(result),    // (I've passed the result to all three, but...)
        getThese(result),
        getThose(result)
    ]))
    .then(() => {
        let state = getState();
        dispatch({
           //dispatch
        });
    }).catch(function (err) {
        console.log(err);
    });
}
Side note about getThis: See this question and its answers. No need for new Promise when you already have a promise. I don't know whether that getThis code is correct in terms of Redux, but here's the same code not using new Promise unnecessarily:
export function getThis() {
    return function(dispatch, getState) {
        return axios.get('api/this/' + lienBuild).then(response => dispatch({type: "FETCH_THIS", data: response.data}));
    };
}
or with destructuring and shorthand properties:
export function getThis() {
    return function(dispatch, getState) {
        return axios.get('api/this/' + lienBuild).then(({data}) => dispatch({type: "FETCH_THIS", data}));
    };
}
or if you can use async/await:
export function getThis() {
    return async function(dispatch, getState) {
        const {data} = await axios.get('api/this/' + lienBuild);
        return dispatch({type: "FETCH_THIS", data});
    };
}