Update your validateForm to destructure the password fields from the form onSubmit event. Create an errors object to track what field errors there are. If the errors object is empty then validation passes, otherwise set error state.

const validateForm = event => {
  event.preventDefault();

  const { confirmPassword, newPassword } = event.target;
  const errors = {};

  const regex = /^(?=.*[a-z])(?=.*[A-Z])(?=.*[^a-zA-Z0-9]).{8,}$/;

  if (!regex.test(newPassword.value)) {
    errors.requirementUnmet = "New password does not meet requirements.";
  }

  if (newPassword.value !== confirmPassword.value) {
    errors.passwordMismatch = "Entered passwords do not match.";
  }

  if (!Object.keys(errors).length) {
    alert("password validation successful.");
    setError(null);
    return;
  }

  setError(errors);
};

Attach validateForm to the form's onSubmit handle.

<Form onSubmit={validateForm}>

Ensure you've a type="submit" button in the form.

<Button
  className="mail-submit-btn"
  block
  type="submit"
  value={loading ? "Loading..." : "Login"}
>
  Submit
</Button>

Updated stackblitz (though I've not an account so copying updated code below.)

Note: I simply JSON.stringify(error) the error for simplicity, but you'll want to render this more gracefully.

Entire ResetPassword.js

import React, { useState } from "react";
import {
  Button,
  Form,
  FormGroup,
  Input,
  Col,
  Label,
  Modal,
  ModalHeader,
  ModalBody,
  ModalFooter
} from "reactstrap";

const useFormInput = initialValue => {
  const [value, setValue] = useState(initialValue);

  const handleChange = e => {
    setValue(e.target.value);
  };
  return {
    value,
    onChange: handleChange
  };
};

const ResetPassword = props => {
  // form inputs
  const email = useFormInput("");
  const newPassword = useFormInput("");
  const confirmPassword = useFormInput("");
  // using hooks
  const [loading, setLoading] = useState(false);
  const [error, setError] = useState(null);

  const [modal, setModal] = useState(false);

  const toggle = () => setModal(!modal);

  const validateForm = event => {
    event.preventDefault();

    const { confirmPassword, newPassword } = event.target;
    const errors = {};

    const regex = /^(?=.*[a-z])(?=.*[A-Z])(?=.*[^a-zA-Z0-9]).{8,}$/;

    if (!regex.test(newPassword.value)) {
      errors.requirementUnmet = "New password does not meet requirements.";
    }

    if (newPassword.value !== confirmPassword.value) {
      errors.passwordMismatch = "Entered passwords do not match.";
    }

    if (!Object.keys(errors).length) {
      alert("password validation successful.");
      setError(null);
      return;
    }

    setError(errors);
  };

  const handleReset = e => {
    e.preventDefault();
    setError(null);
    setLoading(true);
    // some unrelated redux code
  };

  return (
    <div className="mail-reset" id="forgotPassword">
      <div className="mail-reset-content">
        <Form onSubmit={validateForm}>
          <h3 className="form-title">Enter Information</h3>

          <FormGroup row>
            <Label for="Email" sm={2}>
              Email
            </Label>
            <Col sm={4}>
              <Input
                type="email"
                name="email"
                id="Email"
                placeholder="Email"
                aria-label="email address"
                aria-describedby="email address"
                aria-invalid="false"
                {...email}
              />
            </Col>
          </FormGroup>
          <FormGroup row>
            <Label for="newPassword" sm={2}>
              New Password
            </Label>
            <Col sm={4}>
              <Input
                type="password"
                name="password"
                id="newPassword"
                placeholder="New Password"
                aria-label="new password"
                aria-describedby="new password"
                aria-invalid="false"
                {...newPassword}
              />
            </Col>
          </FormGroup>
          <FormGroup row>
            <Label for="confirmPassword" sm={2}>
              Confirm Password
            </Label>
            <Col sm={4}>
              <Input
                type="password"
                name="confirmPassword"
                id="confirmPassword"
                placeholder="Confirm Password"
                aria-label="new password"
                aria-describedby="new password"
                aria-invalid="false"
                {...confirmPassword}
              />
            </Col>
          </FormGroup>

          <div className="modal-wrapper">
            <Col sm={{ size: 4, offset: 2 }}>
              <Button onClick={toggle} className="passwordReqBtn">
                Password Requirements
              </Button>
            </Col>
            <Modal isOpen={modal} toggle={toggle} className="mail-reset-modal">
              <ModalHeader toggle={toggle}>Password Requirements</ModalHeader>
              <ModalBody>
                Your password must contain at least one capital letter, one
                number and one lowercase letter, and it must contain at least 8
                characters
              </ModalBody>
              <ModalFooter>
                <Button onClick={toggle} className="btn-modal pull-right">
                  OK
                </Button>{" "}
              </ModalFooter>
            </Modal>
          </div>

          <div className="form-actions">
            {error && (
              <>
                <small style={{ color: "red" }}>{JSON.stringify(error)}</small>
                <br />
              </>
            )}
            <br />

            <Col lg={{ offset: 2 }} sm={{ size: 1 }}>
              <Button
                className="mail-reset-btn"
                block
                type="submit"
                value={loading ? "Loading..." : "Login"}
              >
                Submit
              </Button>
              <Button
                className="mail-reset-btn"
                block
                type="reset"
                value={loading ? "Loading..." : "Login"}
                onClick={handleReset}
              >
                Reset
              </Button>
            </Col>
          </div>
        </Form>
      </div>
    </div>
  );
};

export default ResetPassword;