The lack of such a form is deliberate. Clients often wish to use their own methods for making contracts. Some, for example, will require the contract to be made as a deed, in which case specific rules have to be followed. In other parts of the world other specific laws may apply to the setting up of a contract. The NEC panel therefore left it up to the parties as to how they wish their contracts to be formed. This does not apply to the short contracts, because they are for simple, straightforward work, supply or services, and therefore a simple offer and acceptance type arrangement will normally be appropriate.
It is therefore up to the parties as to how they formalise their contract. It can, if they wish, be as simple as the employer writing and accepting the contractor’s tender. Or it can be by a suitably worded letter signed by both parties, as you suggest. Alternatively it can be something much more formally executed, in which case you may need to get some legal advice. There is some more information on this for example in the ECC’s guidance notes.