Can my foreign company issue invoices to the Estonian company or vice versa?
Yes, you can, but please note that the pricing between these two companies has to correspond to the market conditions in order to avoid corporate income tax consequences as both companies are related to each other through you and transfer pricing regulations apply.
In addition, the Estonian tax authorities may regard such payments as the remuneration of the member of the management board (salary) if the activities are performed solely by you.
If your non-VAT registered Estonian company buys services from outside Estonia, it must apply for a limited VAT number and pay VAT on the services bought. Therefore, if your Estonian company buys services from your foreign company, it is better to apply for regular VAT and then reverse charge the services, rather than just pay VAT on those services.