GDPR shouldn't affect raising signatures for political purposes much, if at all. Signatures and other personal/identifying data can be collected and processed if it is to fulfill a public task (such as to choose candidates for an election), and if the people giving their signatures consent to their political data being processed.
Article 6 of GDPR covers what purposes represent lawful reasons for processing data. Raising signatures and processing collected data is definitely in the public interest as it shows party/candidate support, and as long as it adheres to EU/member state law then it's fine to do so under GDPR.
Article 6, paragraph 1:
Processing shall be lawful only if and to the extent that at least one of the following applies:
(e) processing is necessary for the performance of a task carried out in the public interest or in the exercise of official authority vested in the controller;
And paragraph 3:
The basis for the processing referred to in point (c) and (e) of paragraph 1 shall be laid down by:
(a) Union law; or
(b) Member State law to which the controller is subject.
As for processing particularly sensitive data, such as political/religious beliefs or biometric/health data, Article 9 of GDPR covers these how special categories of personal data should be processed.
Article 9, paragraph 1:
Processing of personal data revealing racial or ethnic origin, political opinions, religious or philosophical beliefs, or trade union membership, and the processing of genetic data, biometric data for the purpose of uniquely identifying a natural person, data concerning health or data concerning a natural person’s sex life or sexual orientation shall be prohibited.
And paragraph 2:
Paragraph 1 shall not apply if one of the following applies:
a) the data subject has given explicit consent to the processing of those personal data for one or more specified purposes, except where Union or Member State law provide that the prohibition referred to in paragraph 1 may not be lifted by the data subject;
In short, Article 6 says that as long as it's fulfilling a purpose in public interest and adheres to any laws that the collector is subject to, then the data is fine to process, and petitions/signature raising can definitely fall in this category. Further, since you would reasonably expect a person signing a political petition to also consent to their political data being used, processing this sensitive data is definitely allowed by Article 9.