I am wanting to write some songs based on the poems by Emily Dickenson. Since she is dead for more than a century, so I need to ask for permission for the texts? If so, who should I contact to?

Out of nothing more than curiosity, I Googled 'copyright Emily Dickenson'.

This brought up a couple of sites that might be able to help, so try it yourself to see if they are useful.

I assume you mean Dickinson. She is almost certainly out of copyright. For safety if you publish or perform, use a disclaimer such as:
Every effort has been made to trace the copyright holders...if any have been missed the publishers will make the necessary arangement at the first opportunity....
ED died 1886, most of her exquisite work was only discovered afterwards. She liked hyphens, so enjoy expressing those in your music !