Material-UI chips - How to put the delete button on the left side of the chip?

There is no out of the box property for that, however you can build your own chip with the icon on the left side of the Chip.

import { withStyles } from '@material-ui/core/styles';
import Typography from '@material-ui/core/Typography';
import PropTypes from 'prop-types';
import React from 'react';
import IconButton from '@material-ui/core/IconButton';
import CancelRoundedIcon from '@material-ui/icons/CancelRounded';


const styles = theme => ({
  root: {
    margin: '4px',
    backgroundColor: '#e0e0e0',
    display: 'inline-flex',
    boxSizing: 'border-box',
    borderRadius: '16px',
  },
  label: {
    padding: 10,
    margin: 0,
  },
  cancelButton: {
    cursor: 'pointer',
    margin: "7px 5px 0 -5px",
    color: 'rgba(0, 0, 0, 0.26)'
  }
});

class CustomChip extends React.Component {
  render() {
    const { classes } = this.props;
    return (
      <div className={classes.root}>
        <Typography variant="body2" gutterBottom className={classes.label}>
          {this.props.label || ''}
        </Typography>

        {this.props.onDelete &&
          <CancelRoundedIcon
            className={classes.cancelButton}
            onClick={() => {
                this.props.onDelete(this.props.label)
            }}
          />
        }
      </div>
    );
  }
}

CustomChip.propTypes = {
  classes: PropTypes.object.isRequired,
};

export default withStyles(styles)(CustomChip);