Churchill used "Constitutional" here as defined by The Oxford Dictionary as sense 1
Relating to an established set of principles governing a state.
This means that the public opinion which guides the Ministers should be that which is expressed by means consistent with the constitution. By implication this would include formal elections, direct communication with elected representatives, properly conducted opinion polls, referendums and so on: even peaceful mass demonstrations can be considered as constitutional means. What would be excluded would be any sort of violent protest, threats, intimidation, blackmail or other illegal activity.
In the past there have been representations made which were not constitutional at the time but are now regarded as being legitimate. Exmples include:
The St Peter's Field demonstration of 1817 in Manchester which was considered illegal by the ruling classes and was violently dispersed leading to the Peterloo Massacre.
The formation of a trade union in Tolpuddle in 1834 for which the participants were transported to Australia.
The actions of the suffragettes in the early 20th century in pursuit of votes for women which were often illegal, sometimes violent and resulted in imprisonment, hardship, and on at least one occasion death for the campaigners.
The reason that these actions are now considered praiseworthy and even heroic is that the consititution of the time did not give power to the people involved (the working class in the first two cases and women in the third). By 1947 all adults had constitutional rights so Churchill could make the statement that he did.
However the UK does not have a written constitution, its constituion consists of a combination of laws, conventions and legal opinions so deciding what is "constitutional" in the UK is more difficult than it would be in the US for example.
It is difficult to think of a synonym for constitutional means if Japanese does not have a direct translation. Legal methods or Socially acceptable methods come close but do not quite carry the same implication of compliance with the basis of democratic government.