How to get data from javascript array object / JSON and use it in React JS

First you need to parse the vehicle values into separate arrays of JSX options elements. I prefer to do this with a reduce function...

const vehicles = this.state.veh_data.reduce((acc, veh) => {
      let make = Object.keys(veh)[0],
          vehModels = veh[make];

      return {
            makes: [
                  ...acc.makes,
                  <option value={make}>{make}</option>
            ],
            models:[
                  ...acc.models,
                  ...vehModels.map(model => {
                       return <option value={model}>{model}</option>
                  })
            ]  
       };
},{makes:[], models:[]});

Since each element in your veh_data array is an object you need to access its first (and only) property with Object.keys() to get all keys of the object, then just grab the first (and only) key.

Model can be access by getting the value of that key in each veh object you're iterating over

...

<FormGroup>
      <Input type="select" name="select1" id="select1">
      <option selected disabled>Make</option>
      {vehicles.makes}
      </Input>
</FormGroup>

...

<FormGroup>
  <Input type="select" name="select2" id="select2">
        <option selected disabled>Model</option>
        {vehicles.models}
  </Input>
</FormGroup>

...

Given the context I also don't see a need to store the veh_data to the component state, which is a waste of resources unless that data will be updating the UI dynamically so this would be the complete solution....

https://jsfiddle.net/mfpv1L6s/

(NOTE: I understand this won't give you the full solution you 'should' be looking for as it will show the model options for every vehicle rather than the only the models related to the currently selected 'make', but this should be a good starting point for you.)

EXTRA CREDIT: I'd also parse out the option presentation values to something a little more attractive like this...

keyToOption(key){
        return key.split('-')
                .map(word=> word.slice(0,1).toUpperCase + word.slice(1))
                .join(' ');
 }

Then add it to the reduce function like this...

makes: [
    ...acc.makes,
    <option value={make}>{this.keyToOption(make)}</option>
],

I've included the above in the JSFiddle

(Complete working example....)

import React, { Component } from "react";
import { Form, FormGroup, Input } from "reactstrap";
// import { veh_data } from '../shared/vehicle_make_and_models'
const veh_data = [
  { "alfa-romeo": ["145", "90", "Alfa 6", "Alfasud"] },
  { "aston-martin": ["15", "2-Litre", "AM Vantage", "Atom", "Cygnet", "DB2"] },
  { audi: ["100", "200", "A1", "A2", "A3", "A4", "A5", "A6", "A7"] }
];

const vehicles = parseVehicleList(veh_data)

class ImgAndForm extends Component { 
  state = { modelSelected: null };

  handleSearch(event) {
    alert("Search button clicked");
    event.preventDefault();
  }

  handleModelChange(event) {
    console.log(event.target.value);
    this.setState({ modelSelected: event.target.value });
  }

  render() {
    const selectedModels =
      this.state.modelSelected && this.state.modelSelected.length ? (
        vehicles.models[this.state.modelSelected]
      ) : (
        <option value="">(Please Select Make)</option>
      );

    return (
      <div>
        <header className="headerbg d-flex">
          <div className="container my-auto">
            <div className="row">
              <div className="offset-1 col-10 offset-lg-0 col-lg-4">
                <div id="search-form-div" className="container">
                  <div className="row">
                    <div className="col-12 my-4">
                      <h3>Search</h3>
                      <Form onSubmit={this.handleSearch}>
                        <FormGroup>
                          <Input
                            onChange={e => this.handleModelChange(e)}
                            type="select"
                            name="select1"
                            id="select1"
                          >
                            <option value="">Make</option>
                            {vehicles.makes}
                          </Input>
                        </FormGroup>
                        <FormGroup>
                          <Input type="select" name="select2" id="select2">
                            {selectedModels}
                          </Input>
                        </FormGroup>
                        <FormGroup>
                          <Input type="select" name="select3" id="select3">
                            <option value="">Min Price</option>
                            <option value="500">500</option>
                          </Input>
                        </FormGroup>
                        <FormGroup>
                          <Input type="select" name="select4" id="select4">
                            <option value="">Max Price</option>
                            <option value="2000">2000</option>
                          </Input>
                        </FormGroup>
                        <FormGroup>
                          <Input
                            type="submit"
                            name="search"
                            id="search"
                            className="btn btn-primary"
                            value="Search"
                          />
                        </FormGroup>
                      </Form>
                    </div>
                  </div>
                </div>
              </div>
            </div>
          </div>
        </header>
      </div>
    );
  }
}

export default ImgAndForm;

function parseVehicleList(data) {
  return data.reduce(
    (acc, veh, i) => {
      let make = Object.keys(veh)[0],
        vehModels = veh[make];

      return {
        makes: [
          ...acc.makes,
          <option key={make + i} value={make}>
            {keyToOption(make)}
          </option>
        ],
        models: {
          ...acc.models,
          [make]: vehModels.map((model, i) => {
            return (
              <option key={make + model + i} value={model}>
                {keyToOption(model)}
              </option>
            );
          })
        }
      };
    },
    { makes: [], models: {} }
  );
}

function keyToOption(key) {
  return key
    .split("-")
    .map(word => word.slice(0, 1).toUpperCase() + word.slice(1))
    .join(" ");
}